write the function below in the form y = f(u) and u = g(x). then find \\(\\frac{dy}{dx}\\) as a function of…
write the function below in the form y = f(u) and u = g(x). then find \\(\\frac{dy}{dx}\\) as a function of x.\ny = 7(cos x)^{-7}\nwrite the function in the form y = f(u) and u = g(x). choose the correct answer below.\na. y = 7u^{-7} and u = cos x\nb. y = cos u and u = x\nc. y = u and u = cos x\nd. y = cos u and u = 7x^{-7}
Answer
Explanation:
Step1: Identify f(u) and g(x)
Given $y = 7(\cos x)^{-7}$, if we let $u=\cos x$, then $y = 7u^{-7}$. So $f(u)=7u^{-7}$ and $g(x)=\cos x$.
Step2: Find $\frac{dy}{du}$
Using the power - rule for differentiation $\frac{d}{du}(au^n)=nau^{n - 1}$, for $y = 7u^{-7}$, we have $\frac{dy}{du}=7\times(-7)u^{-7 - 1}=-49u^{-8}$.
Step3: Find $\frac{du}{dx}$
Since $u = \cos x$, then $\frac{du}{dx}=-\sin x$.
Step4: Use the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$
Substitute $\frac{dy}{du}=-49u^{-8}$ and $\frac{du}{dx}=-\sin x$ into the chain - rule. Replace $u$ with $\cos x$, we get $\frac{dy}{dx}=(-49(\cos x)^{-8})\times(-\sin x)=49\frac{\sin x}{\cos^{8}x}$.
For the first part of the question (writing in the form $y = f(u)$ and $u = g(x)$):
Answer:
A. $y = 7u^{-7}$ and $u=\cos x$
For finding $\frac{dy}{dx}$:
Answer:
$\frac{dy}{dx}=49\frac{\sin x}{\cos^{8}x}$