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 f(u) and g(x)

Let $y = f(u)=e^{u}$ and $u = g(x)=1 - 10x$. This is in line with the chain - rule requirements for differentiating composite functions.

Step2: Differentiate y with respect to u and u with respect to x

The derivative of $y = f(u)=e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$. The derivative of $u = g(x)=1 - 10x$ with respect to $x$ is $\frac{du}{dx}=-10$.

Step3: Apply 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. Since $u = 1 - 10x$, we have $\frac{dy}{dx}=e^{u}\cdot(-10)=- 10e^{1 - 10x}$.

Answer:

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