Differentiation：練習問題

Find the gradient of the curve \(y = 4x^3 - 7x^2 + 5\) at the point where \(x = 2\).

A curve has the equation \(y = \frac{1}{3}x^3 - 4x + 7\). Find the coordinates of the two stationary points on this curve.

A curve has the equation \(y = (2x - 3)^4\). Find the value of \(\frac{dy}{dx}\) when \(x = 2\).

Find the gradient of the curve \(y = 4x^2 - 7x + 5\) at the point where \(x = 2\).

A curve has the equation \( y = 2x^2 - 8x + 5 \).

(a) Find the gradient of the curve at the point where \( x = 3 \).

(b) Determine the coordinates of the stationary point on the curve and state its nature.

A curve has the equation \( y = \frac{8}{2x - 1} \).

(a) Find an expression for \( \frac{dy}{dx} \).
(b) A point \( P \) moves along the curve. At the point where \( x = 1.5 \), the \( x \)-coordinate of \( P \) is increasing at a constant rate of \( 0.3 \) units per second. Find the rate of change of the \( y \)-coordinate of \( P \) at this instant.
(c) Find the equation of the normal to the curve at the point where \( x = 2 \), giving your answer in the form \( ay = bx + c \).