Exponential & Log Derivatives — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

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

1. 1.

   Differentiate the exponential or logarithmic function.

   $$f(x)=e^x$$

   Answer: $e^x$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=e^x$$
2. 2.

   Differentiate the exponential or logarithmic function.

   $$f(x)=\log \left(x\right)$$

   Answer: $\frac{1}{x}$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=\frac{1}{x}$$
3. 3.

   Differentiate the exponential or logarithmic function.

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

   Answer: $3e^x$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=3e^x$$
4. 4.

   Differentiate the exponential or logarithmic function.

   $$f(x)=\log \left(x^2+1\right)$$

   Answer: $\frac{2x}{x^2+1}$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=\frac{2x}{x^2+1}$$
5. 5.

   Differentiate the exponential or logarithmic function.

   $$f(x)=e^{x^2}$$

   Answer: $2x{e}^{x^2}$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=2x{e}^{x^2}$$
6. 6.

   Differentiate the exponential or logarithmic function.

   $$f(x)=x\log \left(x\right)$$

   Answer: $\log \left(x\right)+1$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=\log \left(x\right)+1$$
7. 7.

   Differentiate the exponential or logarithmic function.

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

   Answer: $5e^x$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=5e^x$$
8. 8.

   Differentiate the exponential or logarithmic function.

   $$f(x)=2\log \left(x\right)$$

   Answer: $\frac{2}{x}$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=\frac{2}{x}$$
9. 9.

   Differentiate the exponential or logarithmic function.

   $$f(x)=e^{3x}$$

   Answer: $3e^{3x}$

   Step-by-step solution

   1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
   2. $$f^{\prime }(x)=3e^{3x}$$
10. 10.

    Differentiate the exponential or logarithmic function.

    $$f(x)=\log \left(2x+1\right)$$

    Answer: $\frac{2}{2x+1}$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=\frac{2}{2x+1}$$
11. 11.

    Differentiate the exponential or logarithmic function.

    $$f(x)=x+e^x$$

    Answer: $e^x+1$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=e^x+1$$
12. 12.

    Differentiate the exponential or logarithmic function.

    $$f(x)=e^x\sin \left(x\right)$$

    Answer: $\sqrt{2}{e}^x\sin \left(x+\frac{\pi }{4}\right)$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=\sqrt{2}{e}^x\sin \left(x+\frac{\pi }{4}\right)$$
13. 13.

    Differentiate the exponential or logarithmic function.

    $$f(x)=\log \left(x^3\right)$$

    Answer: $\frac{3}{x}$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=\frac{3}{x}$$
14. 14.

    Differentiate the exponential or logarithmic function.

    $$f(x)=x^2{e}^x$$

    Answer: $x\left(x+2\right)e^x$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=x\left(x+2\right)e^x$$
15. 15.

    Differentiate the exponential or logarithmic function.

    $$f(x)=e^{-x}$$

    Answer: $-e^{-x}$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=-e^{-x}$$
16. 16.

    Differentiate the exponential or logarithmic function.

    $$f(x)=\frac{\log \left(x\right)}{x}$$

    Answer: $\frac{1-\log \left(x\right)}{x^2}$

    Step-by-step solution

    1. $$(e^u{)}^{\prime }=e^u{u}^{\prime },(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$$
    2. $$f^{\prime }(x)=\frac{1-\log \left(x\right)}{x^2}$$

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• Limits at Infinity
• Continuity
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• Power Rule
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• 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