find f(3) if f(x) = \\frac{x^{5}}{15}+5x^{2}.\nf(3) = \n(simplify your answer.)
find f(3) if f(x) = \\frac{x^{5}}{15}+5x^{2}.\nf(3) = \n(simplify your answer.)
Answer
Explanation:
Step1: Find the derivative of $f(x)$
Use the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $y=\frac{x^{5}}{15}$, its derivative is $\frac{1}{15}\times5x^{4}=\frac{x^{4}}{3}$. For $y = 5x^{2}$, its derivative is $5\times2x=10x$. So $f'(x)=\frac{x^{4}}{3}+10x$.
Step2: Evaluate $f'(x)$ at $x = 3$
Substitute $x = 3$ into $f'(x)$. $f'(3)=\frac{3^{4}}{3}+10\times3$. First, calculate $\frac{3^{4}}{3}=\frac{81}{3}=27$. Second, $10\times3 = 30$. Then $f'(3)=27 + 30$.
Answer:
$57$