Question

Differentiate the following functions with respect to x:
(i) ln(1 + sin^(2) x)
(ii) x^x
.

Answer 1

(i) $\dfrac{d}{dx} \ln(1+\sin^2x) = \dfrac{1}{1+\sin^2x}\cdot\dfrac{d}{dx}(1+\sin^2x) = \dfrac{1}{1+\sin^2x}\cdot (0+2\sin x\cdot\cos x) = \dfrac{\sin2x}{1+\sin^2x}.$

(ii)

$\dfrac{d}{dx} x^x = \dfrac{d}{dx} e^{x\ln x} = e^{x\ln x}\cdot \dfrac{d}{dx} (x\ln x) = e^{x\ln x}\cdot (\ln x + \dfrac{x}{x}) = e^{x\ln x}\cdot (1+\ln x) = \\ = x^x\cdot(1+\ln x).$