Differentiation Of Exponential and Logarithmic Functions

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Step 1: Use the rule \( \frac{d}{dx}(e^{u}) = e^{u} \cdot \frac{du}{dx} \) Step 2: Here, \( u = 3x \), so \( \frac{du}{dx} = 3 \) Step 3: \( \frac{dy}{dx} = e^{3x} \cdot 3 = 3e^{3x} \) Answer: \( \…

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Step 1: Let \( u = x^2 \), so \( \frac{du}{dx} = 2x \) Step 2: Use \( \frac{d}{dx}(e^u) = e^u \cdot \frac{du}{dx} \) Step 3: \( \frac{dy}{dx} = e^{x^2} \cdot 2x = 2x e^{x^2} \) Answer: \( \frac{dy}…

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Step 1: Use \( \frac{d}{dx}(\log u) = \frac{1}{u} \cdot \frac{du}{dx} \) Step 2: \( u = 5x + 2 \), so \( \frac{du}{dx} = 5 \) Step 3: \( \frac{dy}{dx} = \frac{1}{5x+2} \cdot 5 = \frac{5}{5x+2} \) A…

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Step 1: Let \( u = \sin x \), so \( \frac{du}{dx} = \cos x \) Step 2: \( \frac{d}{dx}(\log u) = \frac{1}{u} \cdot \frac{du}{dx} \) Step 3: \( \frac{dy}{dx} = \frac{1}{\sin x} \cdot \cos x = \cot x \…

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Step 1: Let \( u = \sqrt{x} = x^{1/2} \), so \( \frac{du}{dx} = \frac{1}{2\sqrt{x}} \) Step 2: \( \frac{dy}{dx} = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}} \) Answer: \( \frac{dy}{dx} = \frac{e^{\sqrt{…

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Step 1: Let \( u = \log x \), so \( \frac{du}{dx} = \frac{1}{x} \) Step 2: \( \frac{d}{dx}(\log u) = \frac{1}{u} \cdot \frac{du}{dx} \) Step 3: \( \frac{dy}{dx} = \frac{1}{\log x} \cdot \frac{1}{x} …

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Step 1: Use product rule: \( (uv)' = u'v + uv' \) Let \( u = x \), \( v = e^x \) Step 2: \( u' = 1 \), \( v' = e^x \) Step 3: \( \frac{dy}{dx} = 1 \cdot e^x + x \cdot e^x = e^x(1 + x) \) Answer: \…

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Step 1: Use quotient rule: \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \) Let \( u = e^x \), \( v = x \) Step 2: \( u' = e^x \), \( v' = 1 \) Step 3: \( \frac{dy}{dx} = \frac{e^x \cdot x -…