Differentiation Of Exponential and Logarithmic Functions

True

Yes, by chain rule: derivative of \( x^2 \) is \( 2x \), so \( \frac{d}{dx} e^{x^2} = 2x e^{x^2} \).

5e^{5x+2}

Using chain rule: derivative of \( 5x+2 \) is 5, so derivative is \( 5e^{5x+2} \).

\frac{2}{x}

Using chain rule: derivative of \( \log(x^2) = \frac{1}{x^2} \times 2x = \frac{2}{x} \). Alternatively, \( \log(x^2) = 2\log x \), derivative is \( \frac{2}{x} \).

True

Yes, because \( \log(\sqrt{x}) = \frac{1}{2}\log x \), so derivative is \( \frac{1}{2} \cdot \frac{1}{x} = \frac{1}{2x} \).