標準函數的求導：练习题

Find the derivative \( \frac{dy}{dx} \) of the function \( y = e^{3x} + \ln(x) \).

Find the gradient of the normal to the curve \( y = \ln(x^2 + 1) \) at the point where \( x = 1 \).<\/p>

Consider the function \( f(x) = a^x \), where \( a > 0 \).
Given that the gradient of the curve at the point where it crosses the \( y \)-axis is \( \ln(5) \), find the value of \( f'(1) \).<\/p>

Find the gradient of the tangent to the curve \( y = 3\sin(2x) + 4\cos(3x) \) at the point where \( x = \pi \).

Given the function \( f(x) = 5e^{2x} - 4\ln(3x) \), find the exact value of \( f'(\frac{1}{2}) \).

A curve has the equation \( y = 3e^{2x} + 2\ln(x^2 + 1) \).
(a) Find the derivative \( \frac{dy}{dx} \) of the function using standard rules of differentiation.
(b) Find the exact gradient of the tangent to the curve at the point where \( x = 1 \).
(c) Show that the second derivative is given by \( \frac{d^2y}{dx^2} = 12e^{2x} + \frac{4(1 - x^2)}{(x^2 + 1)^2} \).

A curve has the equation \( y = f(x) \) where \( f(x) = \frac{\sin(x)}{2 + \cos(x)} \) for the interval \( 0 \le x \le 2\pi \).
(a) Use the quotient rule to show that \( f'(x) = \frac{1 + 2\cos(x)}{(2 + \cos(x))^2} \).
(b) Find the \( x \)-coordinates of the stationary points of the curve in the given interval, giving your answers in terms of \( \pi \).
(c) Determine the set of values of \( x \) for which \( f(x) \) is a decreasing function.
(d) Find the equation of the tangent to the curve at the point where \( x = \frac{\pi}{2} \). Give your answer in the form \( y = mx + c \).