The chain rule: แบบฝึกหัด

Find the derivative \( \frac{dy}{dx} \) for the function \( y = \cos(e^x) \).

Consider the function \( f(x) = \sin^{-1}(2x) \). Find the value of \( f''(x) \) at \( x = \frac{1}{4} \).

Let \( f(x) = x^3 + x \). Let \( g(x) = f^{-1}(x) \) be the inverse function of \( f \). Find the value of \( g'(2) \).

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

Given the equation \( x^2 + y^2 = 25 \), find the slope of the tangent line at the point \( (3, 4) \) using implicit differentiation.

A curve is defined by the implicit equation \( x^2 + 2xy - y^3 = 7 \). Determine the equation of the tangent line to the curve at the point \( (3, 2) \).