Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] y = tan(Οx) (g(x), f(u)) = ___________ Find the derivative dy/dx. dy/dx = ___________
Accepted Solution
A:
Answer:The inner function is [tex]u = \pi x[/tex].The outer function is [tex]f(u) = \tan{u}[/tex].[tex]\frac{dy}{dx} = y^{\prime} = \sec^{2}(u)*\pi = \pi\sec^{2}{\pi x}[/tex]Step-by-step explanation:The inner function is the one we apply the outer function to.So[tex]y = \tan{\pi x}[/tex]We apply the outer function tangent to [tex]\pi x[/tex].So, the inner function is [tex]u = \pi x[/tex].The outer function is [tex]f(u) = \tan{u}[/tex].The derivative of a compositve function[tex]y = f(u)[/tex] in which [tex]u = g(x)[/tex] is given by the following function.[tex]y^{\prime} = f^{\prime}(u)*g^{\prime}(x)[/tex]So[tex]f(u) = \tan{u}[/tex][tex]f^{\prime}(u) = \sec^{2}{u}[/tex][tex]u = g(x) = \pi x[/tex][tex]g^{\prime}(x) = \pi[/tex]So[tex]y^{\prime} = f^{\prime}(u)*g^{\prime}(x)[/tex][tex]\frac{dy}{dx} = y^{\prime} = \sec^{2}(u)*\pi = \pi\sec^{2}{\pi x}[/tex]