differentiate the function. y=(8 - x)^{50} \\frac{dy}{dx}=\\square
differentiate the function. y=(8 - x)^{50} \\frac{dy}{dx}=\\square
Answer
Explanation:
Step1: Identify the outer - inner functions
Let $u = 8 - x$, then $y = u^{50}$.
Step2: Differentiate the outer function
The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=50u^{49}$ using the power rule $\frac{d}{du}(u^n)=nu^{n - 1}$.
Step3: Differentiate the inner function
The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=-1$.
Step4: Apply the chain - rule
By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=50u^{49}$ and $\frac{du}{dx}=-1$ into the chain - rule formula. Since $u = 8 - x$, we have $\frac{dy}{dx}=50(8 - x)^{49}\cdot(-1)$.
Answer:
$-50(8 - x)^{49}$