Differentiation：練習問題

Given that \( y = (5x - 2)^4 \), use the chain rule to find \( \frac{dy}{dx} \).

A function is given by \( f(x) = \frac{2x + 1}{x - 2} \). Use the quotient rule to find \( f'(x) \).

A function is defined by \( f(x) = 4x^3 - 5x + 2 \). Find the derivative \( f'(x) \).

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

Find the value of \( \frac{dy}{dx} \) at the point \( (1, 1) \) for the curve defined by the implicit equation \( x^2 + y^2 + 3xy = 5 \).

Given that \(f(x) = x^4 - 2x^2 + 7\), find the value of the derivative \(f'(x)\) when \(x = 2\).

Find the equation of the tangent to the curve \( y = x^2 + 4x + 1 \) at the point where the curve intersects the \( y \)-axis.

Find the equation of the tangent to the curve \( y = \frac{4x - 1}{2x + 3} \) at the point where it crosses the \( y \)-axis. Give your answer in the form \( ax + by + c = 0 \).

A curve has the equation \( y = 3x^2 + 5x - 2 \).
(a) Find the gradient of the curve at the point where \( x = 1 \).
(b) Find the equation of the tangent to the curve at the point \( (1, 6) \), giving your answer in the form \( y = mx + c \).

A curve is defined by the equation \( x^2 y + 2y^2 = 12 \).
(a) Use implicit differentiation to find an expression for \( \frac{dy}{dx} \) in terms of \( x \) and \( y \).
(b) Find the gradient of the curve at the point \( (2, 2) \).
(c) Find the coordinates of the two points on the curve where the tangent is horizontal.