Differentiation：練習問題

Find the derivative of the function \( f(x) = 4x^3 - 2x^2 + 7x - 5 \) with respect to \( x \).

Find the x-coordinate of the local minimum point of the curve \( y = x^3 - 3x^2 - 9x + 5 \).

The radius \( r \) of a circle is increasing at a constant rate of \( 0.5 \text{ cm s}^{-1} \). Find the rate of increase of the area \( A \) of the circle at the instant when the radius is \( 10 \text{ cm} \).

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

Using the chain rule, find the derivative \( \frac{dy}{dx} \) for the function \( y = \sqrt{4x^2 + 9} \).

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

Find the derivative \(\frac{dy}{dx}\) of the function \(y = \sqrt{2x^3 + 5}\) using the chain rule.

Find the coordinates of the stationary point on the curve \(y = x^2 + \frac{16}{x}\) for \(x > 0\) and determine its nature using the second derivative.

A curve has the equation \( y = 2x^3 - 9x^2 + 12x - 5 \).

(a) Find an expression for \( \frac{dy}{dx} \).
(b) Find the \( x \)-coordinates of the stationary points on the curve.
(c) Use the second derivative \( \frac{d^2y}{dx^2} \) to determine the nature of each stationary point.