Techniques of differentiation：練習問題

A curve has the equation \( y = e^{3x} \cos(2x) \). Which of the following expressions represents the derivative \( \frac{dy}{dx} \)?

A curve is defined by the equation \( y = \frac{\ln(x)}{x^2} \) for \( x > 0 \). Determine the exact \( y \)-coordinate of the stationary point and identify its nature.

The function \( f(x) \) is defined by \( f(x) = e^{2x} \sin(3x) \).
Find the derivative \( f'(x) \).<\/p>

The function \( f \) is defined by \( f(x) = x^3 + 2x - 1 \). Given that \( g \) is the inverse function of \( f \), find the exact value of the derivative \( g'(2) \).

Consider the function \( f(x) = \frac{\ln(x)}{2x + 5} \) for \( x > 0 \). Use the quotient rule to find an expression for \( f'(x) \).

Find the derivative of the function \( f(x) = \frac{e^{3x}}{x + 1} \) with respect to \( x \) using the quotient rule, and simplify your result.

A spherical balloon is being inflated such that its volume \( V \) increases at a constant rate of \( 10\text{ cm}^3\text{s}^{-1} \).
Find the rate of increase of the surface area \( A \) of the balloon at the instant when the radius is \( 5\text{ cm} \).

The function \( f \) is defined by \( f(x) = x \sin(2x) \).
Use the product rule to find the derivative \( f'(x) \) and hence determine the exact value of \( f'\left(\frac{\pi}{4}\right) \).

A curve is defined by the function \( f(x) = \frac{x+2}{\sqrt{2x+5}} \) for \( x > -2.5 \).
(a) Use the quotient rule to show that the derivative is given by \( f'(x) = \frac{x+3}{(2x+5)^{3/2}} \).
(b) Find the equation of the tangent to the curve at the point where \( x = 2 \). Give your answer in the form \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are integers.
(c) Determine whether the function \( f \) is increasing or decreasing at the point where \( x = -1 \). Justify your answer using the sign of \( f'(x) \).

A function is defined as \( f(x) = e^{-x} \tan(x) \) for the domain \( -\frac{\pi}{2} < x < \frac{\pi}{2} \).
(a) Use the product rule to show that the first derivative is \( f'(x) = e^{-x} (\sec^2 x - \tan x) \).
(b) Find the exact value of \( f'\left(\frac{\pi}{4}\right) \).
(c) By differentiating \( f'(x) \) with respect to \( x \), show that the second derivative is given by \( f''(x) = e^{-x} (2\sec^2 x \tan x - 2\sec^2 x + \tan x) \).
(d) Hence, determine the exact value of \( f''\left(\frac{\pi}{4}\right) \).