Differentiation：練習問題

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

The curve \(y = \frac{1}{3}x^3 - x^2 - 3x + 5\) has two stationary points. Find the \(x\)-coordinate of the local maximum point.

A closed cylindrical container has a fixed volume \(V = 54\pi\) cm³. The total surface area \(S\) is given by \(S = 2\pi r^2 + 2\pi rh\), where \(r\) is the radius and \(h\) is the height. Find the value of \(r\) that minimizes the total surface area.

A stationary point on a curve \(y = f(x)\) occurs at \(x = c\). If \(f'(c) = 0\) and \(f''(c) > 0\), what is the nature of this stationary point?

Find the derivative of the function \(f(x) = \ln(3x^2 + 1)\).

Find the gradient of the tangent to the curve \( y = e^{2x} \) at the point where \( x = 0 \).

Find the coordinates of the stationary point of the curve \( y = x^2 - 4x + 7 \) and determine its nature.

Find the equation of the tangent to the curve \( y = \ln(x^2 + 1) \) at the point where \( x = 1 \).

Find the derivative of the function \(f(x) = e^{2x^2 - 5x + 1}\) with respect to \(x\).

The curve \(C\) has the equation \(y = 2x - e^x\).
(a) Find the \(x\)-coordinate of the stationary point on \(C\), leaving your answer in terms of natural logarithms.
(b) Determine the nature of this stationary point using the second derivative test.