WA5. Derivatives of Inverse Functions, Exponential and Logarithmic Functions¶
Introduction¶
The logarithmic differentiation is defined as follows:
\[
\begin{aligned}
f(x) &= g(x)^{h(x)} \\
\ln(f(x)) &= \ln(g(x)^{h(x)}) \text{ ---(1)}\\
\ln(f(x)) &= h(x) \ln(g(x)) \text{ ---(2)} \\
\frac{f'(x)}{f(x)} &= h'(x) \ln(g(x)) + \frac{h(x) g'(x)}{g(x)} \text{ ---(3)}\\
f'(x) &= f(x) \left( h'(x) \ln(g(x)) + \frac{h(x) g'(x)}{g(x)} \right) \text{ ---(4)} \\
\end{aligned}
\]
Question 1¶
1. Find the derivative for the function \(f(x)=2e^x-8^x\).
\[
\begin{aligned}
f(x) &= 2e^x-8^x \\
\ln(f(x)) &= \ln(2e^x-8^x) \\
\ln(f(x)) &= \ln(2e^x) - \ln(8^x) \\
\ln(f(x)) &= \ln(2) + \ln(e^x) - \ln(8^x) \\
\ln(f(x)) &= \ln(2) + x \ln(e) - x \ln(8) \\
\ln(f(x)) &= \ln(2) + x - x \ln(8) \\
\frac{f'(x)}{f(x)} &= 0 + 1 - \ln(8) \\
f'(x) &= f(x) \left( 1 - \ln(8) \right) \\
f'(x) &= (2e^x-8^x) \left( 1 - \ln(8) \right) \\
\end{aligned}
\]
Question 2¶
2. Find the derivative for the function \(f(z)=z^5-e^z lnz\).
\[
\begin{aligned}
f(z) &= z^5-e^z lnz \\
\ln(f(z)) &= \ln(z^5-e^z lnz) \\
\ln(f(z)) &= \ln(z^5) - \ln(e^z lnz) \\
\ln(f(z)) &= 5 \ln(z) - \ln(e^z) - \ln(lnz) \\
\ln(f(z)) &= 5 \ln(z) - z - \ln(lnz) \\
\frac{f'(z)}{f(z)} &= \frac{5}{z} - 1 - \frac{1}{lnz} \cdot \frac{1}{z} \\
f'(z) &= f(z) \left( \frac{5}{z} - 1 - \frac{1}{lnz} \cdot \frac{1}{z} \right) \\
f'(z) &= \left( z^5-e^z lnz \right) \left( \frac{5}{z} - 1 - \frac{1}{lnz} \cdot \frac{1}{z} \right) \\
\end{aligned}
\]
Question 3¶
3. Find the tangent line to \(f(x)=7^x+4e^x at x=0\).
\[
\begin{aligned}
f(x) &= 7^x+4e^x \\
f'(x) &= 7^x \ln(7) + 4e^x \\
f'(0) &= 7^0 \ln(7) + 4e^0 \\
f'(0) &= 1 \ln(7) + 4 \\
f'(0) &= \ln(7) + 4 \\
f(0) &= 7^0 + 4e^0 \\
f(0) &= 1 + 4 \\
f(0) &= 5 \\
\end{aligned}
\]
\[
\begin{aligned}
y &= f'(0) (x-0) + f(0) \\
y &= (\ln(7) + 4) x + 5 \\
y &= x \ln(7) + 4x + 5 \\
\end{aligned}
\]
Question 4¶
4. Determine if \(G(z)=(z-6)ln z\) is increasing or decreasing at the following points (a) z=1 (b) z=5 © z=20.
\[
\begin{aligned}
G(z) &= (z-6)ln z \\
G'(z) &= (z-6) \frac{1}{z} + ln z \\
G'(1) &= (1-6) \frac{1}{1} + ln 1 \\
G'(1) &= -5 + 0 \\
G'(1) &= -5 \\
G'(5) &= (5-6) \frac{1}{5} + ln 5 \\
G'(5) &= -1 \frac{1}{5} + ln 5 \\
G'(5) &= -\frac{1}{5} + ln 5 \\
G'(5) &= -0.2 + 1.609 \\
G'(5) &= 1.409 \\
G'(20) &= (20-6) \frac{1}{20} + ln 20 \\
G'(20) &= 14 \frac{1}{20} + ln 20 \\
G'(20) &= 0.7 + 2.995 \\
G'(20) &= 3.695 \\
\end{aligned}
\]
Question 5¶
5. Find the derivative for the function \(f(x)=(x+1)^x\).
\[
\begin{aligned}
f(x) &= (x+1)^x \\
\ln(f(x)) &= \ln((x+1)^x) \\
\ln(f(x)) &= x \ln(x+1) \\
\frac{f'(x)}{f(x)} &= \ln(x+1) + \frac{x}{x+1} \\
f'(x) &= f(x) \left( \ln(x+1) + \frac{x}{x+1} \right) \\
f'(x) &= (x+1)^x \left( \ln(x+1) + \frac{x}{x+1} \right) \\
\end{aligned}
\]
Question 6¶
6. Find the derivative for the function \(f(x)=(x)^{x+1}\).
\[
\begin{aligned}
f(x) &= (x)^{x+1} \\
\ln(f(x)) &= \ln((x)^{x+1}) \\
\ln(f(x)) &= (x+1) \ln(x) \\
\frac{f'(x)}{f(x)} &= \ln(x) + \frac{x+1}{x} \\
f'(x) &= f(x) \left( \ln(x) + \frac{x+1}{x} \right) \\
f'(x) &= (x)^{x+1} \left( \ln(x) + \frac{x+1}{x} \right) \\
\end{aligned}
\]
Question 7¶
7. Find the derivative for the function \(f(x)=(\sqrt{x})^x\).
\[
\begin{aligned}
f(x) &= (\sqrt{x})^x \\
\ln(f(x)) &= \ln((\sqrt{x})^x) \\
\ln(f(x)) &= x \ln(\sqrt{x}) \\
\ln(f(x)) &= x \frac{1}{2} \ln(x) \\
\frac{f'(x)}{f(x)} &= \frac{1}{2} \ln(x) + \frac{x}{2x} \\
f'(x) &= f(x) \left( \frac{1}{2} \ln(x) + \frac{x}{2x} \right) \\
f'(x) &= (\sqrt{x})^x \left( \frac{1}{2} \ln(x) + \frac{x}{2x} \right) \\
f'(x) &= (\sqrt{x})^x \left( \frac{1}{2} \ln(x) + \frac{1}{2} \right) \\
\end{aligned}
\]
Question 8¶
8. Find \(\frac{dy}{dx}\) for \(\sqrt{3x^2+1} (3x^4+1)^3\).
\[
\begin{aligned}
f(x) &= \sqrt{3x^2+1} (3x^4+1)^3 \\
\ln(f(x)) &= \ln(\sqrt{3x^2+1} (3x^4+1)^3) \\
\ln(f(x)) &= \ln(\sqrt{3x^2+1}) + \ln((3x^4+1)^3) \\
\ln(f(x)) &= \frac{1}{2} \ln(3x^2+1) + 3 \ln((3x^4+1)) \\
\frac{f'(x)}{f(x)} &= \frac{1}{2} \frac{6x}{3x^2+1} + 3 \frac{12x^3}{3x^4+1} \\
f'(x) &= f(x) \left( \frac{1}{2} \frac{6x}{3x^2+1} + 3 \frac{12x^3}{3x^4+1} \right) \\
f'(x) &= \sqrt{3x^2+1} (3x^4+1)^3 \left( \frac{1}{2} \frac{6x}{3x^2+1} + 3 \frac{12x^3}{3x^4+1} \right) \\
\end{aligned}
\]
Question 9¶
9. Find \(\frac{dy}{dx}\) for \(y=3x^{3x}\).
\[
\begin{aligned}
f(x) &= 3x^{3x} \\
\ln(f(x)) &= \ln(3x^{3x}) \\
\ln(f(x)) &= \ln(3) + \ln(x^{3x}) \\
\ln(f(x)) &= \ln(3) + 3x \ln(x) \\
\frac{f'(x)}{f(x)} &= 0 + 3 \ln(x) + 3x \frac{1}{x} \\
f'(x) &= f(x) \left( 3 \ln(x) + 3x \frac{1}{x} \right) \\
f'(x) &= 3x^{3x} \left( 3 \ln(x) + 3x \frac{1}{x} \right) \\
f'(x) &= 3x^{3x} \left( 3 \ln(x) + 3 \right) \\
\end{aligned}
\]