Chapter 3, Problem 1SYU

To determine

To find if the below statement is true or false.

“If $f\left(x\right)=5x^2+1$, then $f^{\prime }\left(-1\right)=-10$ .”

Answer to Problem 1SYU

The statement is true.

Explanation of Solution

Given:

Statement as “If $f\left(x\right)=5x^2+1$, then $f^{\prime }\left(-1\right)=-10$ .”

Formula Used:

Formula of derivatives are as follows:

$\frac{d}{dx}\left(c\right)=0$ ,where $c$ is a constant.

$\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

$\frac{d}{dx}c.f\left(x\right)=c.\frac{d}{dx}f\left(x\right)$

Calculation:

Function is given as

$f\left(x\right)=5x^2+1$.

Derivative of the above function with respect to $x$ is calculated as

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(5x^2+1\right)$

$f^{\prime }\left(x\right)=\frac{d}{dx}\left(5x^2\right)+\frac{d}{dx}\left(1\right)$

$f^{\prime }\left(x\right)=5.\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)$

$f^{\prime }\left(x\right)=5.\left(2\times x^{2-1}\right)+\left(0\right)$

$f^{\prime }\left(x\right)=10x$

Now, we need to find $f^{\prime }\left(-1\right)$.

Substituting $x=-1$ in $f^{\prime }\left(x\right)=10x$, we get

$f^{\prime }\left(-1\right)=10\left(-1\right)$

$f^{\prime }\left(-1\right)=-10$.

Conclusion:

Hence, the statement is true.