Question Type 7: Learning to apply all the rules, product, chain and quotient together for any derivatives of composite functions of x Exercises

Differentiate $f(x) = \dfrac{x^2\,\sec\big(\sqrt{\ln x}\big)}{1 + x e^{-x}}$ for $x>1$.

Differentiate $f(x) = \dfrac{x^2 e^{\sin x}}{\sqrt{1+x^2}}$.

Differentiate $f(x) = \dfrac{(1+\ln x)e^{x^2}}{\cos x}$ for $x>0$.

Differentiate $f(x) = \dfrac{\sin(\ln(x^2+1))}{x e^{x}}$.

Differentiate $f(x) = \dfrac{e^{\cos x}\,\sin x}{x^2}$.

Differentiate $f(x) = \dfrac{e^{-x}\cos(\ln x)}{x\,\ln x}$ for $x>0$, $x \ne 1$.

Differentiate $f(x) = \dfrac{(\ln x)^3}{e^{x}\cos(\ln x)}$ for $x>0$ by rewriting as a product.

Differentiate $f(x) = \ln\left(\dfrac{\sin(x^3)}{x e^{x}}\right)$ for $x > 0$.

Differentiate $f(x) = \dfrac{x\,\arctan(\sqrt{x})}{\ln x}$ for $x>0$, $x\ne 1$.

Differentiate $f(x) = (\ln x)^2 \tan(x^2)$ for $x > 0$.

Differentiate $f(x) = \dfrac{\sqrt{\ln(1+x^2)}}{x\sin x}$ for $x>0$ and $x\ne k\pi$ where $k\in\mathbb{Z}$.

Differentiate $f(x) = \left(x + \frac{1}{x}\right) e^{-\ln(\cos x)}$ on intervals where defined.