The chain rule：练习题

If \(f(x) = \ln(\sin(x))\) for \(0 < x < \pi\), what is \(f'(x)\)?

Find the value of \(\frac{dy}{dx}\) at the point \((2, 1)\) for the curve defined by the equation \(x^3 - 2xy + y^3 = 5\).

Let \( f(x) = x^3 + 2x + 1 \). If \( g(x) \) is the inverse function of \( f(x) \), find the value of \( g'(4) \).

Given the curve defined by the equation \( x^2 y + y^3 = 10 \), find the value of \( \frac{dy}{dx} \) at the point \( (1, 2) \).

Find the slope of the normal line to the curve given parametrically by \(x = t^3 + 1\) and \(y = t^2 - t\) at the point where \(t = 2\).

Find the derivative of the function \( g(x) = \ln(3x + 1) \) with respect to \( x \).

Let \( f(x) = e^{2x} \cos(3x) \). Find the instantaneous rate of change of \( f(x) \) at \( x = 0 \).

A curve is defined by the equation \( x + y = \arctan(x^2 y) \). Determine the value of \( \frac{dy}{dx} \) at the point \( (0, 0) \).

A curve is defined by the implicit equation \( x^2 + \sin(y) = xy \).
(a) Use implicit differentiation to find an expression for \( \frac{dy}{dx} \) in terms of \( x \) and \( y \).
(b) Find the slope of the tangent line to the curve at the point \( (0, \pi) \).
(c) Let \( h(x) = f(g(x)) \). Given that \( g(1) = 0 \), \( g'(1) = 2 \), and that at the point \( (0, \pi) \) on the curve, \( y \) can be expressed as a function of \( x \) (i.e., \( f(0) = \pi \)), find the value of \( h'(1) \).

A curve is defined by the implicit equation \( e^{xy} + y^2 = 2x + 1 \).
(a) Find an expression for \( \frac{dy}{dx} \) in terms of \( x \) and \( y \).
(b) Determine the equation of the tangent line to the curve at the point \( (0, 0) \).