Power Rule — Practice Problems and Worked Solutions

The formulas you need here

26 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Differentiate with respect to x.

   $$f(x)=3x^4$$

   Answer: $12x^3$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }3\cdot 4=12$$
   2. $$\text{Step the exponent down by one: }{x}^3$$
   3. $$f^{\prime }(x)=12x^3$$
2. 2.

   Differentiate with respect to x.

   $$f(x)=5x^2$$

   Answer: $10x$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }5\cdot 2=10$$
   2. $$\text{Step the exponent down by one: }{x}^1$$
   3. $$f^{\prime }(x)=10x$$
3. 3.

   Differentiate with respect to x.

   $$f(x)=2x^5$$

   Answer: $10x^4$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }2\cdot 5=10$$
   2. $$\text{Step the exponent down by one: }{x}^4$$
   3. $$f^{\prime }(x)=10x^4$$
4. 4.

   Differentiate with respect to x.

   $$f(x)=7x^3$$

   Answer: $21x^2$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }7\cdot 3=21$$
   2. $$\text{Step the exponent down by one: }{x}^2$$
   3. $$f^{\prime }(x)=21x^2$$
5. 5.

   Differentiate with respect to x.

   $$f(x)=4x^6$$

   Answer: $24x^5$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }4\cdot 6=24$$
   2. $$\text{Step the exponent down by one: }{x}^5$$
   3. $$f^{\prime }(x)=24x^5$$
6. 6.

   Differentiate with respect to x.

   $$f(x)=6x^2$$

   Answer: $12x$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }6\cdot 2=12$$
   2. $$\text{Step the exponent down by one: }{x}^1$$
   3. $$f^{\prime }(x)=12x$$
7. 7.

   Differentiate with respect to x.

   $$f(x)=2x^3$$

   Answer: $6x^2$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }2\cdot 3=6$$
   2. $$\text{Step the exponent down by one: }{x}^2$$
   3. $$f^{\prime }(x)=6x^2$$
8. 8.

   Differentiate with respect to x.

   $$f(x)=3x^2$$

   Answer: $6x$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }3\cdot 2=6$$
   2. $$\text{Step the exponent down by one: }{x}^1$$
   3. $$f^{\prime }(x)=6x$$
9. 9.

   Differentiate with respect to x.

   $$f(x)=4x^3$$

   Answer: $12x^2$

   Step-by-step solution

   1. $$\text{Bring the exponent down and multiply: }4\cdot 3=12$$
   2. $$\text{Step the exponent down by one: }{x}^2$$
   3. $$f^{\prime }(x)=12x^2$$
10. 10.

    Differentiate with respect to x.

    $$f(x)=5x^4$$

    Answer: $20x^3$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }5\cdot 4=20$$
    2. $$\text{Step the exponent down by one: }{x}^3$$
    3. $$f^{\prime }(x)=20x^3$$
11. 11.

    Differentiate with respect to x.

    $$f(x)=2x^7$$

    Answer: $14x^6$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }2\cdot 7=14$$
    2. $$\text{Step the exponent down by one: }{x}^6$$
    3. $$f^{\prime }(x)=14x^6$$
12. 12.

    Differentiate with respect to x.

    $$f(x)=8x^2$$

    Answer: $16x$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }8\cdot 2=16$$
    2. $$\text{Step the exponent down by one: }{x}^1$$
    3. $$f^{\prime }(x)=16x$$
13. 13.

    Differentiate with respect to x.

    $$f(x)=3x^5$$

    Answer: $15x^4$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }3\cdot 5=15$$
    2. $$\text{Step the exponent down by one: }{x}^4$$
    3. $$f^{\prime }(x)=15x^4$$
14. 14.

    Differentiate with respect to x.

    $$f(x)=6x^3$$

    Answer: $18x^2$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }6\cdot 3=18$$
    2. $$\text{Step the exponent down by one: }{x}^2$$
    3. $$f^{\prime }(x)=18x^2$$
15. 15.

    Differentiate with respect to x.

    $$f(x)=9x^2$$

    Answer: $18x$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }9\cdot 2=18$$
    2. $$\text{Step the exponent down by one: }{x}^1$$
    3. $$f^{\prime }(x)=18x$$
16. 16.

    Differentiate with respect to x.

    $$f(x)=2x^6$$

    Answer: $12x^5$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }2\cdot 6=12$$
    2. $$\text{Step the exponent down by one: }{x}^5$$
    3. $$f^{\prime }(x)=12x^5$$
17. 17.

    Differentiate with respect to x.

    $$f(x)=5x^3$$

    Answer: $15x^2$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }5\cdot 3=15$$
    2. $$\text{Step the exponent down by one: }{x}^2$$
    3. $$f^{\prime }(x)=15x^2$$
18. 18.

    Differentiate with respect to x.

    $$f(x)=4x^4$$

    Answer: $16x^3$

    Step-by-step solution

    1. $$\text{Bring the exponent down and multiply: }4\cdot 4=16$$
    2. $$\text{Step the exponent down by one: }{x}^3$$
    3. $$f^{\prime }(x)=16x^3$$
19. 19.

    Differentiate with respect to x.

    $$f(x)=2x^3+5x$$

    Answer: $6x^2+5$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=6x^2+5$$
20. 20.

    Differentiate with respect to x.

    $$f(x)=3x^3+4x^2$$

    Answer: $x\left(9x+8\right)$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=9x^2+8x$$
21. 21.

    Differentiate with respect to x.

    $$f(x)=3x^4+2x^2$$

    Answer: $12x^3+4x$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=12x^3+4x$$
22. 22.

    Differentiate with respect to x.

    $$f(x)=5x^2+7x$$

    Answer: $10x+7$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=10x+7$$
23. 23.

    Differentiate with respect to x.

    $$f(x)=2x^5+3x^2$$

    Answer: $10x^4+6x$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=10x^4+6x$$
24. 24.

    Differentiate with respect to x.

    $$f(x)=6x^3+4x$$

    Answer: $18x^2+4$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=18x^2+4$$
25. 25.

    Differentiate with respect to x.

    $$f(x)=3x^3+5x^2$$

    Answer: $x\left(9x+10\right)$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=9x^2+10x$$
26. 26.

    Differentiate with respect to x.

    $$f(x)=2x^4+7x^2$$

    Answer: $8x^3+14x$

    Step-by-step solution

    1. $$\text{Differentiate each term with the power rule.}$$
    2. $$f^{\prime }(x)=8x^3+14x$$

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• Limits at Infinity
• Continuity
• Derivative from First Principles
• Trigonometric Derivatives
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Related Rates
• Curve Analysis
• Optimization
• Antiderivatives
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution