Problem 1 - Calculus Early Transcendentals

Question

Two equivalent forms of the Chain Rule for calculating the derivative of \(y=f(g(x))\) are presented in this section. State both forms.

Step-by-Step Solution

Answer

Question: State the two equivalent forms of the Chain Rule used to find the derivative of a composite function y = f(g(x)). Answer: The two equivalent forms of the Chain Rule are: 1. dy/dx = (dy/du) * (du/dx), where y = f(u) and u = g(x). 2. (dy/dx) = f'(g(x)) * g'(x), where y = f(g(x)).

Step 1: Form 1: Chain Rule using dy/dx, dy/du, and dx/du

The first form of the Chain Rule uses the derivatives dy/dx, dy/du, and du/dx. If y = f(u) and u = g(x), then the derivative dy/dx can be calculated as: dy/dx = (dy/du) * (du/dx) So, we first find the derivative dy/du (with respect to u) and du/dx (with respect to x), and then multiply them together to obtain the overall derivative dy/dx.

Step 2: Form 2: Chain Rule using f' and g' notation

The second form of the Chain Rule uses the prime notation f'(x) and g'(x) for the derivatives of the functions f and g. If y = f(g(x)), then the derivative dy/dx can be calculated as: (dy/dx) = f'(g(x)) * g'(x) In this form, we first find the derivatives f'(x) and g'(x), evaluate f'(g(x)), and then multiply the results together to obtain the overall derivative dy/dx.