Question

In Exercises 63–68, find a function that has the given derivative and value. In each case you can find the answer with an educated guess-and-check process while thinking about the

chain rule.

$f\text{'}\left(x\right)=5{\left(x^2+1\right)}^4\left(2x\right),f\left(1\right)=25$

Short answer

The function is$f\left(x\right)={\left(x^2+1\right)}^5-7$

Step-by-step solution

Step 1. Given Information

The given derivative is$f\text{'}\left(x\right)=5{\left(x^2+1\right)}^4\left(2x\right),f\left(1\right)=25$

Step 2. Finding antiderivative

$$\begin{gathered} f\left(x\right)=\int 5{\left(x^2+1\right)}^4\left(2x\right)dx \\ f\left(x\right)=\int 5t^4\left(2x\right)\frac{dt}{2x}=5\int t^{4}dt \\ f\left(x\right)=5\frac{t^5}{5}+C \\ f\left(x\right)=t^5+C \\ f\left(x\right)={\left(x^2+1\right)}^5+C \end{gathered}$$

Step 3. Finding the value of C

Given that $f\left(1\right)=25$

$$\begin{gathered} {\left(1+1\right)}^5+C=25 \\ 32+C=25 \\ C=25-32=-7 \end{gathered}$$

The function become$f\left(x\right)={\left(x^2+1\right)}^5-7$