Differentiation：練習問題

Find the gradient of the tangent to the curve \(y = \frac{x^2+3}{2x-1}\) at the point where \(x=1\).

A curve is defined by the parametric equations:
\( x = e^t \)
\( y = e^{2t} - 4e^t \)
Find the coordinates of the stationary point on the curve.

Given that \(y = \sqrt{4x^2+9}\), find the value of \(\frac{dy}{dx}\) when \(x=2\).

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

Find the gradient of the curve defined by the equation \(x^2 + 2xy + y^3 = 4\) at the point \((1, 1)\).

Find the derivative of the function \(y = 4x^3 - 5\sqrt{x} + 7\) with respect to \(x\).

Find \(\frac{dy}{dx}\) for the function \(y = x^2 e^{3x}\), expressing your answer in terms of \(x\).

The volume \(V\) of a sphere is increasing at a constant rate of \(0.1 \text{ cm}^3 \text{ s}^{-1}\). Find the rate at which the radius \(r\) is increasing when the radius is \(2\) cm, given that \(V = \frac{4}{3}\pi r^3\).

A curve has the equation \(y = 2x^3 + 3x^2 - 12x + 5\).
(a) Find an expression for \(\frac{dy}{dx}\).
(b) Find the \(x\)-coordinates of the two stationary points on the curve.
(c) Determine the nature of each stationary point using the second derivative test.

A curve is defined by the parametric equations \(x = 2e^t\) and \(y = e^{2t} - 4t\).
(a) Find an expression for \(\frac{dy}{dx}\) in terms of \(t\).
(b) Find the exact \(x\)-coordinate of the stationary point of the curve.
(c) By considering the sign of \(\frac{dy}{dx}\) on either side of the stationary point, or by using the second derivative, show that this stationary point is a minimum.