📝 Unit 5: Exponential & Trig Derivatives
\(\frac{d}{dx}e^x,\ b^x,\ \sin x,\ \cos x,\ \tan x\) · Logarithmic differentiation
🔄 Not Graded
Purpose: Self-check derivatives of transcendental functions.
Score: 0 / 12
Question 1
\(\frac{d}{dx}(e^{3x})\) at \(x=0\):
Solution:
\(\frac{d}{dx}e^{3x}=3e^{3x}\). At 0: \(3\).
Question 2
If \(f(x)=2^x\), then \(f'(x)\) is:
Solution:
\(\frac{d}{dx}b^x=b^x\ln b\). With \(b=2\): \(2^x\ln 2\).
Question 3
Differentiate \(y=xe^x\) at \(x=1\).
Solution:
Product: \(y'=e^x+xe^x=(1+x)e^x\). At 1: \(2e\approx 5.437\).
Question 4
\(\frac{d}{dx}\sin(2x)\) at \(x=0\):
Solution:
\(2\cos(2x)\); at 0: \(2\).
Question 5
If \(f(x)=\cos(x^2)\), then \(f'(x)=\)
Solution:
Chain: outer \(-\sin\), inner \(2x\): \(-2x\sin(x^2)\).
Question 6
\(\frac{d}{dx}\tan(3x)\) at \(x=0\):
Solution:
\(\frac{d}{dx}\tan u=\sec^2 u\cdot u'\); \(3\sec^2(3x)\); at 0: \(3\cdot 1=3\).
Question 7
Differentiate \(y=e^{\sin x}\). Value of \(y'\) at \(x=0\):
Solution:
\(y'=e^{\sin x}\cos x\); at 0: \(1\cdot 1=1\).
Question 8
\(\frac{d}{dx}\!\left(\sin^2 x\right)\) at \(x=\pi/4\):
Solution:
\(2\sin x\cos x=\sin(2x)\); at \(\pi/4\): \(\sin(\pi/2)=1\).
Question 9
If \(y=\ln(x^2+1)\), find \(y'(2)\).
Solution:
\(y'=\frac{2x}{x^2+1}\); at 2: \(4/5=0.8\).
Question 10
Use logarithmic differentiation to find \(\frac{dy}{dx}\) for \(y=x^x\) at \(x=1\).
Solution:
\(\ln y=x\ln x\); \(y'/y=\ln x+1\); \(y'=x^x(\ln x+1)\). At 1: \(1\cdot(0+1)=1\).
Question 11
\(\frac{d}{dx}\!\left(\sin x\cos x\right)\) at \(x=\pi/6\):
Solution:
\(\cos^2 x-\sin^2 x=\cos(2x)\); at \(\pi/6\): \(\cos(\pi/3)=1/2\).
Question 12
For \(y=e^{-x}\sin x\), \(y'(0)\) is:
Solution:
\(y'=-e^{-x}\sin x+e^{-x}\cos x=e^{-x}(\cos x-\sin x)\); at 0: \(1\cdot 1=1\).
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