Derivative worksheet

1. Derivative f″(x): 8cos(4x) − 16x·sin(4x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x·sin(4x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x·sin(4x)] = sin(4x) + 4x·cos(4x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x] = 1}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(4x)] = 4cos(4x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [4x] = 4}$

6. Simplify first, so the next round is easier

   $\text{f′(x) = sin(4x) + 4x·cos(4x)}$

7. Take the derivative of that result to get the second derivative

   $\text{f″(x) = d/dx [sin(4x) + 4x·cos(4x)]}$

8. Sum rule: differentiate each term separately and add the results

   $\text{d/dx [sin(4x) + 4x·cos(4x)] = 8cos(4x) − 16x·sin(4x)}$

9. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(4x)] = 4cos(4x)}$

10. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [4x] = 4}$

11. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [4x·cos(4x)] = 4(cos(4x) − 4x·sin(4x))}$

12. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x·cos(4x)] = cos(4x) − 4x·sin(4x)}$

13. Chain rule with d/du cos(u) = −sin(u): multiply by the derivative of the inside

    $\text{d/dx [cos(4x)] = −4sin(4x)}$

14. …then the same kinds of rules 1 more time, on the smaller pieces

15. Tidy up: combine like terms and simplify

    $\text{f″(x) = 8cos(4x) − 16x·sin(4x)}$

16. Put x = 2 into f″(x) to get its value at that point

    $\text{f″(}2\text{)}$$=\mathrm{-32.8234641624}$

2. Derivative f‴(x): 36cos(6x) − 216x·sin(6x) − 216x^2·cos(6x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x^2·sin(6x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x^2·sin(6x)] = 2x·sin(6x) + 6x^2·cos(6x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x^2] = 2x}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(6x)] = 6cos(6x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [6x] = 6}$

6. Simplify first, so the next round is easier

   $\text{f′(x) = 2x·sin(6x) + 6x^2·cos(6x)}$

7. Take the derivative of that result to get the second derivative

   $\text{f″(x) = d/dx [2x·sin(6x) + 6x^2·cos(6x)]}$

8. Sum rule: differentiate each term separately and add the results

   $\text{d/dx [2x·sin(6x) + 6x^2·cos(6x)] = 2sin(6x) + 24x·cos(6x) − 36x^2·sin(6x)}$

9. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [2x·sin(6x)] = 2(sin(6x) + 6x·cos(6x))}$

10. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x·sin(6x)] = sin(6x) + 6x·cos(6x)}$

11. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

    $\text{d/dx [sin(6x)] = 6cos(6x)}$

12. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [6x] = 6}$

13. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [6x^2·cos(6x)] = 6(2x·cos(6x) − 6x^2·sin(6x))}$

14. …then the same kinds of rules 4 more times, on the smaller pieces

15. Simplify first, so the next round is easier

    $\text{f″(x) = 2sin(6x) + 24x·cos(6x) − 36x^2·sin(6x)}$

16. Take the derivative of that result to get the third derivative

    $\text{f‴(x) = d/dx [2sin(6x) + 24x·cos(6x) − 36x^2·sin(6x)]}$

17. Sum rule: differentiate each term separately and add the results

    $\text{d/dx [2sin(6x) + 24x·cos(6x) − 36x^2·sin(6x)] = 36cos(6x) − 216x·sin(6x) − 216x^2·cos(6x)}$

18. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [2sin(6x)] = 12cos(6x)}$

19. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

    $\text{d/dx [sin(6x)] = 6cos(6x)}$

20. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [6x] = 6}$

21. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [24x·cos(6x)] = 24(cos(6x) − 6x·sin(6x))}$

22. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x·cos(6x)] = cos(6x) − 6x·sin(6x)}$

23. …then the same kinds of rules 8 more times, on the smaller pieces

24. Tidy up: combine like terms and simplify

    $\text{f‴(x) = 36cos(6x) − 216x·sin(6x) − 216x^2·cos(6x)}$

25. Put x = 1 into f‴(x) to get its value at that point

    $\text{f‴(}1\text{)}$$=\mathrm{-112.4769039861}$

3. Derivative f′(x): sin(3x) + 3x·cos(3x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x·sin(3x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x·sin(3x)] = sin(3x) + 3x·cos(3x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x] = 1}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(3x)] = 3cos(3x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [3x] = 3}$

6. Tidy up: combine like terms and simplify

   $\text{f′(x) = sin(3x) + 3x·cos(3x)}$

7. Put x = 1 into f′(x) to get its value at that point

   $\text{f′(}1\text{)}$$=\mathrm{-2.8288574817}$

8. The point on the curve: f(1)

   $\text{f(}1\text{)}$$=0.1411200081$

9. The tangent line has the same slope at that point: y = f′(a)(x − a) + f(a), written as y = slope × x + intercept

   $\text{y = −2.828857x + 2.969977}$

4. Derivative f″(x): 6x·sin(5x) + 30x^2·cos(5x) − 25x^3·sin(5x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x^3·sin(5x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x^3·sin(5x)] = 3x^2·sin(5x) + 5x^3·cos(5x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x^3] = 3x^2}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(5x)] = 5cos(5x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [5x] = 5}$

6. Simplify first, so the next round is easier

   $\text{f′(x) = 3x^2·sin(5x) + 5x^3·cos(5x)}$

7. Take the derivative of that result to get the second derivative

   $\text{f″(x) = d/dx [3x^2·sin(5x) + 5x^3·cos(5x)]}$

8. Sum rule: differentiate each term separately and add the results

   $\text{d/dx [3x^2·sin(5x) + 5x^3·cos(5x)] = 6x·sin(5x) + 30x^2·cos(5x) − 25x^3·sin(5x)}$

9. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [3x^2·sin(5x)] = 3(2x·sin(5x) + 5x^2·cos(5x))}$

10. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x^2·sin(5x)] = 2x·sin(5x) + 5x^2·cos(5x)}$

11. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

    $\text{d/dx [x^2] = 2x}$

12. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

    $\text{d/dx [sin(5x)] = 5cos(5x)}$

13. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [5x] = 5}$

14. …then the same kinds of rules 5 more times, on the smaller pieces

15. Tidy up: combine like terms and simplify

    $\text{f″(x) = 6x·sin(5x) + 30x^2·cos(5x) − 25x^3·sin(5x)}$

16. Put x = 1 into f″(x) to get its value at that point

    $\text{f″(}1\text{)}$$=26.7294267825$

5. Derivative f‴(x): −75sin(5x) − 125x·cos(5x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x·sin(5x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x·sin(5x)] = sin(5x) + 5x·cos(5x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x] = 1}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(5x)] = 5cos(5x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [5x] = 5}$

6. Simplify first, so the next round is easier

   $\text{f′(x) = sin(5x) + 5x·cos(5x)}$

7. Take the derivative of that result to get the second derivative

   $\text{f″(x) = d/dx [sin(5x) + 5x·cos(5x)]}$

8. Sum rule: differentiate each term separately and add the results

   $\text{d/dx [sin(5x) + 5x·cos(5x)] = 10cos(5x) − 25x·sin(5x)}$

9. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(5x)] = 5cos(5x)}$

10. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [5x] = 5}$

11. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [5x·cos(5x)] = 5(cos(5x) − 5x·sin(5x))}$

12. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x·cos(5x)] = cos(5x) − 5x·sin(5x)}$

13. Chain rule with d/du cos(u) = −sin(u): multiply by the derivative of the inside

    $\text{d/dx [cos(5x)] = −5sin(5x)}$

14. …then the same kinds of rules 1 more time, on the smaller pieces

15. Simplify first, so the next round is easier

    $\text{f″(x) = 10cos(5x) − 25x·sin(5x)}$

16. Take the derivative of that result to get the third derivative

    $\text{f‴(x) = d/dx [10cos(5x) − 25x·sin(5x)]}$

17. Sum rule: differentiate each term separately and add the results

    $\text{d/dx [10cos(5x) − 25x·sin(5x)] = −75sin(5x) − 125x·cos(5x)}$

18. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [10cos(5x)] = −50sin(5x)}$

19. Chain rule with d/du cos(u) = −sin(u): multiply by the derivative of the inside

    $\text{d/dx [cos(5x)] = −5sin(5x)}$

20. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [5x] = 5}$

21. Constant multiple rule: the minus sign stays in front

    $\text{d/dx [−25x·sin(5x)] = −25sin(5x) − 125x·cos(5x)}$

22. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [25x·sin(5x)] = 25(sin(5x) + 5x·cos(5x))}$

23. …then the same kinds of rules 3 more times, on the smaller pieces

24. Tidy up: combine like terms and simplify

    $\text{f‴(x) = −75sin(5x) − 125x·cos(5x)}$

25. Put x = 1 into f‴(x) to get its value at that point

    $\text{f‴(}1\text{)}$$=36.4615474168$

6. Derivative f′(x): 3x^2·sin(4x) + 4x^3·cos(4x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x^3·sin(4x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x^3·sin(4x)] = 3x^2·sin(4x) + 4x^3·cos(4x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x^3] = 3x^2}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(4x)] = 4cos(4x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [4x] = 4}$

6. Tidy up: combine like terms and simplify

   $\text{f′(x) = 3x^2·sin(4x) + 4x^3·cos(4x)}$

7. Put x = 1 into f′(x) to get its value at that point

   $\text{f′(}1\text{)}$$=\mathrm{-4.8849819694}$

8. The point on the curve: f(1)

   $\text{f(}1\text{)}$$=\mathrm{-0.7568024953}$

9. The tangent line has the same slope at that point: y = f′(a)(x − a) + f(a), written as y = slope × x + intercept

   $\text{y = −4.884982x + 4.128179}$

7. Derivative f‴(x): −108sin(6x) − 216x·cos(6x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x·sin(6x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x·sin(6x)] = sin(6x) + 6x·cos(6x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x] = 1}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(6x)] = 6cos(6x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [6x] = 6}$

6. Simplify first, so the next round is easier

   $\text{f′(x) = sin(6x) + 6x·cos(6x)}$

7. Take the derivative of that result to get the second derivative

   $\text{f″(x) = d/dx [sin(6x) + 6x·cos(6x)]}$

8. Sum rule: differentiate each term separately and add the results

   $\text{d/dx [sin(6x) + 6x·cos(6x)] = 12cos(6x) − 36x·sin(6x)}$

9. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(6x)] = 6cos(6x)}$

10. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [6x] = 6}$

11. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [6x·cos(6x)] = 6(cos(6x) − 6x·sin(6x))}$

12. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x·cos(6x)] = cos(6x) − 6x·sin(6x)}$

13. Chain rule with d/du cos(u) = −sin(u): multiply by the derivative of the inside

    $\text{d/dx [cos(6x)] = −6sin(6x)}$

14. …then the same kinds of rules 1 more time, on the smaller pieces

15. Simplify first, so the next round is easier

    $\text{f″(x) = 12cos(6x) − 36x·sin(6x)}$

16. Take the derivative of that result to get the third derivative

    $\text{f‴(x) = d/dx [12cos(6x) − 36x·sin(6x)]}$

17. Sum rule: differentiate each term separately and add the results

    $\text{d/dx [12cos(6x) − 36x·sin(6x)] = −108sin(6x) − 216x·cos(6x)}$

18. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [12cos(6x)] = −72sin(6x)}$

19. Chain rule with d/du cos(u) = −sin(u): multiply by the derivative of the inside

    $\text{d/dx [cos(6x)] = −6sin(6x)}$

20. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [6x] = 6}$

21. Constant multiple rule: the minus sign stays in front

    $\text{d/dx [−36x·sin(6x)] = −36sin(6x) − 216x·cos(6x)}$

22. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [36x·sin(6x)] = 36(sin(6x) + 6x·cos(6x))}$

23. …then the same kinds of rules 3 more times, on the smaller pieces

24. Tidy up: combine like terms and simplify

    $\text{f‴(x) = −108sin(6x) − 216x·cos(6x)}$

25. Put x = 1 into f‴(x) to get its value at that point

    $\text{f‴(}1\text{)}$$=\mathrm{-177.219908111}$

8. Derivative f‴(x): 24x·sin(5x) + 180x^2·cos(5x) − 300x^3·sin(5x) − 125x^4·cos(5x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x^4·sin(5x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x^4·sin(5x)] = 4x^3·sin(5x) + 5x^4·cos(5x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x^4] = 4x^3}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(5x)] = 5cos(5x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [5x] = 5}$

6. Simplify first, so the next round is easier

   $\text{f′(x) = 4x^3·sin(5x) + 5x^4·cos(5x)}$

7. Take the derivative of that result to get the second derivative

   $\text{f″(x) = d/dx [4x^3·sin(5x) + 5x^4·cos(5x)]}$

8. Sum rule: differentiate each term separately and add the results

   $\text{d/dx [4x^3·sin(5x) + 5x^4·cos(5x)] = 12x^2·sin(5x) + 40x^3·cos(5x) − 25x^4·sin(5x)}$

9. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [4x^3·sin(5x)] = 4(3x^2·sin(5x) + 5x^3·cos(5x))}$

10. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x^3·sin(5x)] = 3x^2·sin(5x) + 5x^3·cos(5x)}$

11. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

    $\text{d/dx [x^3] = 3x^2}$

12. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

    $\text{d/dx [sin(5x)] = 5cos(5x)}$

13. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [5x] = 5}$

14. …then the same kinds of rules 5 more times, on the smaller pieces

15. Simplify first, so the next round is easier

    $\text{f″(x) = 12x^2·sin(5x) + 40x^3·cos(5x) − 25x^4·sin(5x)}$

16. Take the derivative of that result to get the third derivative

    $\text{f‴(x) = d/dx [12x^2·sin(5x) + 40x^3·cos(5x) − 25x^4·sin(5x)]}$

17. Sum rule: differentiate each term separately and add the results

    $\text{d/dx [12x^2·sin(5x) + 40x^3·cos(5x) − 25x^4·sin(5x)] = 24x·sin(5x) + 180x^2·cos(5x) − 300x^3·sin(5x) − 125x^4·cos(5x)}$

18. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [12x^2·sin(5x)] = 12(2x·sin(5x) + 5x^2·cos(5x))}$

19. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

    $\text{d/dx [x^2·sin(5x)] = 2x·sin(5x) + 5x^2·cos(5x)}$

20. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

    $\text{d/dx [x^2] = 2x}$

21. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

    $\text{d/dx [sin(5x)] = 5cos(5x)}$

22. Constant multiple rule: keep the constant factor and differentiate the rest

    $\text{d/dx [5x] = 5}$

23. …then the same kinds of rules 11 more times, on the smaller pieces

24. Tidy up: combine like terms and simplify

    $\text{f‴(x) = 24x·sin(5x) + 180x^2·cos(5x) − 300x^3·sin(5x) − 125x^4·cos(5x)}$

25. Put x = 3 into f‴(x) to get its value at that point

    $\text{f‴(}3\text{)}$$=\mathrm{1,240.6349180829}$

9. Derivative f′(x): 4x^3·sin(2x) + 2x^4·cos(2x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x^4·sin(2x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x^4·sin(2x)] = 4x^3·sin(2x) + 2x^4·cos(2x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x^4] = 4x^3}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(2x)] = 2cos(2x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [2x] = 2}$

6. Tidy up: combine like terms and simplify

   $\text{f′(x) = 4x^3·sin(2x) + 2x^4·cos(2x)}$

7. Put x = 2 into f′(x) to get its value at that point

   $\text{f′(}2\text{)}$$=\mathrm{-45.1342757175}$

8. The point on the curve: f(2)

   $\text{f(}2\text{)}$$=\mathrm{-12.1088399249}$

9. The tangent line has the same slope at that point: y = f′(a)(x − a) + f(a), written as y = slope × x + intercept

   $\text{y = −45.134276x + 78.159712}$

10. Derivative f′(x): 3x^2·sin(6x) + 6x^3·cos(6x)

1. First, read the function in the usual order of operations. x is the variable, and angles are in radians

   $\text{f(x) = x^3·sin(6x)}$

2. Product rule, for two parts multiplied together: (uv)′ = u′v + uv′

   $\text{d/dx [x^3·sin(6x)] = 3x^2·sin(6x) + 6x^3·cos(6x)}$

3. Power rule: bring the power down in front and lower it by 1, (x^n)′ = n·x^(n − 1)

   $\text{d/dx [x^3] = 3x^2}$

4. Chain rule with d/du sin(u) = cos(u): multiply by the derivative of the inside

   $\text{d/dx [sin(6x)] = 6cos(6x)}$

5. Constant multiple rule: keep the constant factor and differentiate the rest

   $\text{d/dx [6x] = 6}$

6. Tidy up: combine like terms and simplify

   $\text{f′(x) = 3x^2·sin(6x) + 6x^3·cos(6x)}$

7. Put x = 1 into f′(x) to get its value at that point

   $\text{f′(}1\text{)}$$=4.9227752253$

8. The point on the curve: f(1)

   $\text{f(}1\text{)}$$=\mathrm{-0.2794154982}$

9. The tangent line has the same slope at that point: y = f′(a)(x − a) + f(a), written as y = slope × x + intercept

   $\text{y = 4.922775x − 5.202191}$