Question

find f(x), f(x), and f(x) for the following function. f(x)=4x^{2}+7e^{x} f(x)=□

Explanation:

Step1: Differentiate term - by - term

Use the power rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$ for $4x^2$ and $\frac{d}{dx}(e^x)=e^x$ for $7e^x$.
For $y = 4x^2$, $\frac{d}{dx}(4x^2)=2\times4x^{2 - 1}=8x$. For $y = 7e^x$, $\frac{d}{dx}(7e^x)=7e^x$.
So, $f'(x)=8x + 7e^x$.

Step2: Differentiate $f'(x)$ to get $f''(x)$

Differentiate $8x$ and $7e^x$ in $f'(x)$ term - by - term.
For $y = 8x$, $\frac{d}{dx}(8x)=8$. For $y = 7e^x$, $\frac{d}{dx}(7e^x)=7e^x$.
So, $f''(x)=8 + 7e^x$.

Step3: Differentiate $f''(x)$ to get $f'''(x)$

Differentiate $8$ and $7e^x$ in $f''(x)$ term - by - term.
For $y = 8$, $\frac{d}{dx}(8)=0$. For $y = 7e^x$, $\frac{d}{dx}(7e^x)=7e^x$.
So, $f'''(x)=7e^x$.

Answer:

$f'(x)=8x + 7e^x$, $f''(x)=8 + 7e^x$, $f'''(x)=7e^x$