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

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

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

Find the derivative of the function $f(x) = x \sin(\sqrt{x})$.

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

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

Compute $\dfrac{d}{dx}\left(\dfrac{\cos(\ln x)}{x}\right)$.

Find $\frac{\text{d}}{\text{d}x} \left( \frac{\text{e}^x}{1 + \text{e}^x} \right)$.

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

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

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

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

Find $f'(x)$.