The Derivatives of Exponential Function
\(\frac{d}{dx}\left [ e^{x} \right ]=e^{x}\) \(\frac{d}{dx}\left [ e^{u} \right ]=e^{u}\frac{du}{dx}\)
\(\frac{d}{dx}\left [ a^{x} \right ]=a^{x}lna\) \(\frac{d}{dx}\left [ a^{u} \right ]=a^{u}\left ( lna \right )\frac{du}{dx}\)
Example 1
- Find \({y}”\) if \(y=e^{cosx}.\)
▶️Answer/Explanation
Solution
\({y}’=\frac{d}{dx}\left ( e^{cosx} \right )\)
\(=e^{cosx}\frac{d}{dx}\left ( cosx \right )\) \(\frac{d}{dx}\left [ e^{u} \right ]=e^{u}\frac{du}{dx}\)
\(=e^{cosx}\left ( -sinx \right )\)
\({y}”=e^{cosx}\cdot \left ( -cosx \right )+e^{cosx}\left ( -sinx \right )\cdot \left ( -sinx \right )\)
\(=e^{cosx}\left [ -cosx+sin^{2}x \right ]\)
Example 2
Differentiate \(y=3^{\sqrt{x^{2}-x}}.\)
▶️Answer/Explanation
Solution
\(\frac{dy}{dx}=3^{\sqrt{x^{2}-x}}\left ( ln3 \right )\frac{d}{dx}\left ( \sqrt{x^{2}-x} \right )\) \(\frac{d}{dx}\left [ a^{u} \right ]=a^{u}\left ( lna \right )\frac{du}{dx}\)
\(=3^{\sqrt{x^{2}-x}}\left ( ln3 \right )\left ( \frac{1}{2\sqrt{x^{2}-x}} \right )\frac{d}{dx}\left ( x^{2}-x \right )\)
\(=ln3\left ( 3^{\sqrt{x^{2}-x}} \right )\left ( \frac{2x-1}{2\sqrt{x^{2}-x}} \right )\)
The Derivatives of Logarithmic Function
\(\frac{d}{dx}\left ( lnx \right )=\frac{1}{x}\) \(\frac{d}{dx}\left ( log_{a}x \right )=\frac{1}{lna}\cdot \frac{1}{x}\)
\(\frac{d}{dx}\left ( ln u \right )=\frac{1}{u}\frac{du}{dx}\) \(\frac{d}{dx}\left ( log_{a}u \right )=\frac{1}{lna}\cdot \frac{1}{u}\frac{du}{dx}\)
Properties of Logarithms
\(lnxy=lnx+lny\) \(ln\frac{x}{y}=lnx-lny\)
\(lnx^{p}=plnx\) \(e^{lnx}=x\)
Example 4 □ Find \({y}’\) if \(y=\frac{lnx}{x^{2}}.\)
▶️Answer/Explanation
Solution
\(y=\frac{lnx}{x^{2}}=lnx\cdot x^{-2}\)
\({y}’=lnx\frac{d}{dx}x^{-2}+x^{-2}\frac{d}{dx}lnx\) Product Rule
\(=lnx\left ( -2x^{-3} \right )+x^{-2}\cdot \frac{1}{x}\)
\(=\frac{-2lnx}{x^{3}}+\frac{1}{x^{2}}\cdot \frac{1}{x}\)
\(=\frac{-2lnx+1}{x^{3}}\)
Example 5
- Find \({y}’\) if \(y=x^{lnx}.\)
▶️Answer/Explanation
Solution
\(y=x^{lnx}\)
\(lny=ln\left ( x^{lnx} \right )\) Take natural log of both sides.
\(lny=lnx\cdot lnx=\left ( lnx \right )^{2}\) \(lnx^{p}=plnx\)
\(\frac{d}{dx}lny=\frac{d}{dx}\left ( lnx \right )^{2}\)
\(\frac{1}{y}\frac{d}{dx}\left ( y \right )=2lnx\frac{d}{dx}\left ( lnx \right )\) Chain Rule
\(\frac{1}{y}\cdot {y}’=2lnx\cdot \frac{1}{x}\) \(\frac{dy}{dx}={y}’\)
\({y}’=y\left [ 2lnx\cdot \frac{1}{x} \right ]\) Multiply by y on both sides.
\({y}’=x^{lnx}\cdot \frac{2lnx}{x}\)
Exercises – Derivatives of Exponential and Logarithmic Functions
Multiple Choice Questions
1. \(\underset{h\rightarrow 0}{lim}\frac{\frac{1}{2}\left [ ln\left ( e+h \right )-1 \right ]}{h}\) is
(A) \({f}’\left ( 1 \right )\), where \(f\left ( x \right )=ln\sqrt{x}\)
(B) \({f}’\left ( 1 \right )\), where \(f\left ( x \right )=ln\sqrt{x+e}\)
(C) \({f}’\left ( e \right )\), where \(f\left ( x \right )=ln\sqrt{x}\)
(D) \({f}’\left ( e \right )\), where \(f\left ( x \right )=ln\left ( \frac{x}{2} \right )\)
▶️Answer/Explanation
Ans:
1. C
2. If \(f\left ( x \right )=e^{tanx}\), then \({f}’\left ( \frac{\pi }{4} \right )=\)
(A) \(\frac{e}{2}\) (B) e (C) 2e (D) \(\frac{e^{2}}{2}\)
▶️Answer/Explanation
Ans:
2. C
3. If \(y=ln\left ( cosx \right )\), then \({y}’=\)
(A) − tan x (B) tan x (C) −cot x (D) csc x
▶️Answer/Explanation
Ans:
3. A
4. If \(y=x^{x}\), then \({y}’=\)
(A) \(x^{x}lnx\) (B) \(x^{x}\left ( 1+lnx \right )\) (C) \(x^{x}\left ( x+lnx \right )\) (D) \(\frac{x^{x}lnx}{x}\)
▶️Answer/Explanation
Ans:
4. B
5. If \(y=e^{\sqrt{x^{2}+1}}\), then y’=
(A) \(\sqrt{x^{2}+1}e^{\sqrt{x^{2}+1}}\)
(B) \(2x\sqrt{x^{2}+1}e^{\sqrt{x^{2}+1}}\)
(C) \(\frac{e\sqrt{x^{2}+1}}{\sqrt{x^{2}+1}}\)
(D) \(\frac{xe^{\sqrt{x^{2}+1}}}{\sqrt{x^{2}+1}}\)
▶️Answer/Explanation
Ans:
5. D
6. If \(y=\left ( sinx \right )^{1/x}\), then \({y}’=\)
(A) \(\left ( sinx \right )^{\frac{1}{x}}\left [ \frac{ln\left ( sinx \right )}{x} \right ]\)
(B) \(\left ( sinx \right )^{\frac{1}{x}}\left [ \frac{x-ln\left ( sinx \right )}{x^{2}} \right ]\)
(C) \(\left ( sinx \right )^{\frac{1}{x}}\left [ \frac{xsinx-ln\left ( sinx \right )}{x^{2}} \right ]\)
(D) \(\left ( sinx \right )^{\frac{1}{x}}\left [ \frac{xcotx-ln\left ( sinx \right )}{x^{2}} \right ]\)
▶️Answer/Explanation
Ans:
6. D
7. If \(f\left ( x \right )=ln\left [ sec\left ( lnx \right ) \right ]\), then \({f}’\left ( e \right )=\)
(A) \(\frac{cos1}{e}\) (B) \(\frac{sin1}{e}\) (C) \(\frac{tan1}{e}\) (D) \(\frac{cot1}{e}\)
▶️Answer/Explanation
Ans:
7. C
8. If \(y=x^{ln\sqrt{x}}\), then \({y}’=\)
(A) \(\frac{x^{ln\sqrt{x}}lnx}{2x}\)
(B) \(\frac{x^{ln\sqrt{x}}lnx}{x}\)
(C) \(\frac{2x^{ln\sqrt{x}}lnx}{x}\)
(D) \(\frac{x^{ln\sqrt{x}}\left ( 1+lnx \right )}{x}\)
▶️Answer/Explanation
Ans:
8. B
Free Response Questions
9. Let \(f\left ( x \right )=xe^{x}\) and \(f^{\left ( n \right )}\left ( x \right )\) be the nth derivative of f with respect to x . If \(f^{\left ( 10 \right )}\left ( x \right )=\left ( x+n \right )e^{x}\), what is the value of n ?
▶️Answer/Explanation
Ans:
9. 10
10. Let f and h be twice differentiable functions such that \(h\left ( x \right )=e^{f\left ( x \right )}\). If \({h}”\left ( x \right )=e^{f\left ( x \right )}\left [ 1+x^{2} \right ]\), then \({f}’\left ( x \right )=\)
▶️Answer/Explanation
Ans:
10. x