Rates of change and the definition of the derivative：練習問題

Evaluate the derivative of \( f(x) = \frac{2x - 3}{x^2 + 1} \) at \( x = 1 \).

Using the limit definition of the derivative, find \(f'(x)\) for \(f(x) = \sqrt{3x + 1}\).

Find the \(n\)-th derivative of \(f(x) = x e^x\).

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

Find the derivative of \(f(x) = x^3 e^{2x}\) with respect to \(x\).

Calculate the derivative of \( f(x) = 4e^x - \frac{2}{x^3} \) with respect to \( x \).

Using the basic power rule and constant multiple rule, calculate the derivative of the function \( f(x) = \frac{4}{\sqrt{x}} + 5\pi \) with respect to \( x \).

Using the product rule, find the derivative of \( y = x^2 \sin(x) \).

Consider the function defined by \( f(x) = \frac{1}{x+2} \).
(a) Using the formal definition of the derivative as a limit, find \( f'(x) \).
(b) Evaluate \( f'(1) \).

Consider the function \( f(x) = x^4 - 2x + 5 \).
(a) Using the fundamental power rule and linearity of the derivative, find \( f'(x) \).
(b) Find the value of \( x \) such that the instantaneous rate of change of \( f(x) \) is 2.
(c) Determine the coordinates of the point on the graph where the tangent line is horizontal.