微分：练习题

Given that \( y = \tan(e^{2x}) \), find an expression for \( \frac{dy}{dx} \).

A curve is defined by the parametric equations \( x = \cos t \) and \( y = \sin 2t \). Find the gradient of the curve at the point where \( t = \frac{\pi}{6} \).

Find the x-coordinate of the stationary point of the curve \( y = x^2 \ln x \) for \( x > 0 \).

A curve is defined implicitly by the equation \( 2^x + y^2 = xy + 4 \). Find the value of \( \frac{dy}{dx} \) at the point \( (2, 2) \).

The volume of a sphere is increasing at a constant rate of \( 12\pi \text{ cm}^3\text{s}^{-1} \). Find the rate of increase of the surface area of the sphere, in \( \text{cm}^2\text{s}^{-1} \), at the instant when the radius is \( 3 \text{ cm} \).

Find the gradient of the curve with equation \(y = \frac{x^2+9}{x}\) at the point where \(x = 3\).

Find the coordinates of the two stationary points on the curve with equation \(y = x^2 e^{-x}\) and determine the nature of each point.

The curve \(C\) has equation \(y = 2x^3 - 9x^2 + 12x - 5\).
(a) Find \(\frac{dy}{dx}\).
(b) Find the coordinates of the stationary points of \(C\).
(c) Determine the nature of each stationary point by considering the sign of the second derivative \(\frac{d^2y}{dx^2}\) at these points.

A curve is defined parametrically by \(x = t^2 + 1, y = 4t - t^3\).
(a) Find \(\frac{dy}{dx}\) in terms of \(t\).
(b) Find the equation of the normal to the curve at the point where \(t = 1\), giving your answer in the form \(ax + by + c = 0\).