微分：練習題

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

A curve has the equation \(y = (5x^2 - 3)^4\) . Find the value of \(\frac{dy}{dx}\) when \(x = 1\) .

The volume \(V\) of a sphere is increasing at a constant rate of \(12 \text{ cm}^3 \text{ s}^{-1}\). Find the rate at which the surface area \(A\) is increasing when the radius \(r\) is \(4 \text{ cm}\). The formulae for the volume and surface area of a sphere are \(V = \frac{4}{3} \pi r^3\) and \(A = 4 \pi r^2\).

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

A curve \(C\) has the equation \(y = \frac{x^2}{2x-1}\). Find the equation of the normal to the curve \(C\) at the point where \(x=2\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are integers.

Given the function \(f(x) = 3x^4 - 2x + \frac{1}{x^2}\), find the derivative \(f'(x)\).

Find \(\frac{dy}{dx}\) when \(y = \frac{(x-2)(2x+1)}{x}\), ensuring your answer is simplified fully.

A curve has the equation \(y = x^3 - 9x^2 + 15x\). Find the exact coordinates of the stationary points and determine whether each is a local maximum or minimum.

A curve $C$ has the equation $y = x^3 - 6x^2 + 5x + 12$.

Part (a)

Find $\frac{dy}{dx}$.

Part (b)

Find the coordinates of the two stationary points on the curve $C$.

Part (c)

Determine the nature (maximum or minimum) of each stationary point.

Part (d)

Given that $(x-3)$ is a factor of $x^3 - 6x^2 + 5x + 12$, find the coordinates where the curve $C$ crosses the $x$-axis.

A curve C has the equation \(y = \frac{1}{2}x^2 - 8\sqrt{x} + 6\), for \(x > 0\).

Part (a)

Find \(\frac{dy}{dx}\). (2 points)

Part (b)

Find the coordinates of the stationary point of C. (3 points)

Part (c)

Determine the nature of this stationary point, giving a reason for your answer. (2 points)