let y = f(x), where f(x)=x^7 - 6x. find the differential of the function. dy=(7x^6 - 6)dx

Explanation:

Step1: Recall derivative formula

The derivative of $x^n$ is $nx^{n - 1}$ and the derivative of a constant - times a function is the constant times the derivative of the function.

Step2: Differentiate $f(x)$

For $y = f(x)=x^{7}-6x$, using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $\frac{dy}{dx}=\frac{d}{dx}(x^{7})-\frac{d}{dx}(6x)$. The derivative of $x^{7}$ is $7x^{6}$ and the derivative of $6x$ is $6$. So $\frac{dy}{dx}=7x^{6}-6$.

Step3: Find the differential

The differential $dy$ is given by $dy = f^{\prime}(x)dx$. Since $f^{\prime}(x)=7x^{6}-6$, then $dy=(7x^{6}-6)dx$.

Answer:

$dy=(7x^{6}-6)dx$