Question

write the function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x.
y = $e^{1 - 10x}$
choose the correct form of y in terms of u.
a. y = $e^{u}$, where u = 1 - 10x
b. y = $e^{-1 + 10u}$, where u = -x
c. y = $e^{ux}$, where u = 1 - 10
d. y = $u^{1 - 10x}$, where u = e

Explanation:

Step1: Identify the inner - outer functions

Let $y = f(u)=e^{u}$ and $u = g(x)=1 - 10x$. So the correct form of $y$ in terms of $u$ is $y = e^{u}$, where $u = 1 - 10x$ (Option A).

Step2: Find the derivative of $y$ with respect to $u$

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$.

Step3: Find the derivative of $u$ with respect to $x$

The derivative of $u = 1-10x$ with respect to $x$ is $\frac{du}{dx}=- 10$.

Step4: Use the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=-10$ into the chain - rule formula. Then substitute $u = 1 - 10x$ back in. So $\frac{dy}{dx}=e^{u}\cdot(-10)=-10e^{1 - 10x}$.

Answer:

A. $y = e^{u}$, where $u = 1 - 10x$; $\frac{dy}{dx}=-10e^{1 - 10x}$