Differentiation：練習問題

If \(f(x) = 3x^4 - 2x^2 + 5x - 7\), find \(f'(x)\).

Differentiate \(y = x^3 \cos(4x)\) with respect to \(x\).

A curve is defined parametrically by \(x = \cos^2 t\) and \(y = \sin t\). Find \(\frac{dy}{dx}\) in terms of \(t\).

Find the derivative of \(f(x) = 4x^3 - \frac{2}{x^2} + 5\) with respect to \(x\).

Given that \(f(x) = \frac{x^4 - 2x^2 + x}{\sqrt{x}} \), find \(f'(x) \).

Differentiate \(f(x) = 7x^5 - 2\sqrt{x} + \frac{1}{x}\) with respect to \(x\).

Find the gradient of the curve \(y = x^3 - 4x + 1\) at the point where \(x=2\).

A particle moves such that its displacement \(s\) metres at time \(t\) seconds is given by \(s = \frac{1}{3}t^3 - 2t^2 + 3t\). Find the time(s) when the particle is momentarily at rest.

Find \(\frac{dy}{dx}\) for the function \(y = 3x^4 - 2x^{\frac{3}{2}} + \frac{5}{x}\).

A curve \(C\) has the equation \(y = x^3 - 6x^2 + 5x - 1\).

(a) Find \(\frac{dy}{dx}\).

(b) Find the equation of the tangent to the curve \(C\) at the point where \(x=1\).