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]
