Question Type 6: Deriving functions by applying product, quotient and chain rule Exercises

Differentiate the function $f(x) = x^3\,\sin(2x)$.

Compute $\displaystyle f'(x)$ if $f(x) = (3x^2 + 1)^5$.

Find the derivative of $f(x) = e^{3x}\,\cos(x^2)$.

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

Find $\dfrac{d}{dx}\Bigl(\dfrac{e^x}{1+e^x}\Bigr)$.

Differentiate $f(x) = x^2\,\ln(x)$.

Compute the derivative of $f(x) = \sin(\sqrt{x})\,x$.

Differentiate $f(x) = \dfrac{\ln(x)}{x}$.

Find the derivative of $f(x) = e^{-x}\,\ln(\cos x)$.

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

Compute $\dfrac{d}{dx}\Bigl(\dfrac{\cos(\ln x)}{x}\Bigr)$.

Differentiate the composite function $f(x)=\frac{\cos(\ln(x))\,e^{-x}}{x\ln(x)}.$