Question

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(πx) (g(x), f(u)) = ___________ Find the derivative dy/dx. dy/dx = ___________

Answer 1

Answer:

The inner function is $u = \pi x$.

The outer function is $f(u) = \tan{u}$.

$\frac{dy}{dx} = y^{\prime} = \sec^{2}(u)*\pi = \pi\sec^{2}{\pi x}$

Step-by-step explanation:

The inner function is the one we apply the outer function to.

So

$y = \tan{\pi x}$

We apply the outer function tangent to $\pi x$.

So, the inner function is $u = \pi x$.

The outer function is $f(u) = \tan{u}$.

The derivative of a compositve function

$y = f(u)$ in which $u = g(x)$ is given by the following function.

$y^{\prime} = f^{\prime}(u)*g^{\prime}(x)$

So

$f(u) = \tan{u}$

$f^{\prime}(u) = \sec^{2}{u}$

$u = g(x) = \pi x$

$g^{\prime}(x) = \pi$

So

$y^{\prime} = f^{\prime}(u)*g^{\prime}(x)$

$\frac{dy}{dx} = y^{\prime} = \sec^{2}(u)*\pi = \pi\sec^{2}{\pi x}$