Differentiation：練習問題

Given that \( y = (3x - 1)^4 \), find the derivative \( \frac{dy}{dx} \).

A curve has the equation \(y = (2x - 1)^3 - 24x\).
Find the \(x\)-coordinate of the maximum stationary point on this curve.

A curve is defined by the parametric equations:
\(x = 2\sin^2 \theta\)
\(y = 3\cos \theta\)
for \(0 < \theta < \frac{\pi}{2}\). Find the gradient of the normal to the curve at the point where \(\theta = \frac{\pi}{3}\).

Find the gradient of the curve \( y = 2x^3 - 5x^2 + 4 \) at the point where \( x = 2 \).

Using the product rule, differentiate the function \( y = x^2 e^{3x} \) with respect to \( x \).

Calculate the gradient of the curve \(y = (3x - 1)^4\) at the point where \(x = 1\).

A circle has radius \(r\) cm and area \(A\) cm\(^2\). Given that the radius is increasing at a constant rate of \(0.5 \text{ cm s}^{-1}\), find the rate of increase of the area at the instant when \(r = 8\).

The equation of a curve is \(y^3 + 5x^2y = 26\). Use implicit differentiation to find the gradient of the curve at the point \((1, 2)\), giving your answer as a simplified fraction.

The function \(f\) is defined by \(f(x) = 2x^3 - 9x^2 + 12x - 5\) for \(x \in \mathbb{R}\).
(a) Find the coordinates of the two stationary points on the curve \(y = f(x)\).
(b) Determine the nature of each stationary point using the second derivative \(f''(x)\).

The equation of a curve is \(y = \frac{\ln x}{x^2}\) for \(x > 0\).
(a) Show that \(\frac{dy}{dx} = \frac{1 - 2\ln x}{x^3}\).
(b) Find the exact coordinates of the stationary point on the curve.
(c) Determine whether this stationary point is a maximum or a minimum.