Standard Derivatives — practice questions

Differentiate $y = 4x^3 - \dfrac{2}{x} + 5\sqrt{x}$.

Find $f''(x)$ for $f(x) = \dfrac{x^4 + 3x^2}{2x}$, giving your answer in a form with no negative or fractional powers.

The curve $y = ax^2 + \dfrac{b}{x}$ has a horizontal tangent at the point $(2, 6)$. Find the exact values of $a$ and $b$, and determine whether the point is a maximum or a minimum.

Differentiate $y = e^{3x} + \ln(5x)$.

Find the exact coordinates of the stationary point of $y = xe^{-2x}$.

The curve $y = \ln x - kx$, where $k > 0$, has a maximum value of zero. Find the exact value of $k$.

Differentiate $y = 3\sin 4x - \cos(2x + 1)$.

Find all values of $x$ in $0 \leq x \leq 2\pi$ at which the curve $y = 2\sin x + \cos 2x$ has a stationary point.

Show that the curve $y = \tan x - x$ has no maximum or minimum points on the interval $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$, and describe the behaviour of the curve at $x = 0$.