find the derivative of f(x). f(x) = √x^7 write your answer as a constant times a power of x. f(x) =
find the derivative of f(x). f(x) = √x^7 write your answer as a constant times a power of x. f(x) =
Answer
Explanation:
Step1: Rewrite the function
Rewrite $\sqrt{x^{7}}$ as $x^{\frac{7}{2}}$ using the rule $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$. So, $f(x)=x^{\frac{7}{2}}$.
Step2: Apply the power - rule for derivatives
The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. For $f(x)=x^{\frac{7}{2}}$, we have $n=\frac{7}{2}$. So, $f^\prime(x)=\frac{7}{2}x^{\frac{7}{2}-1}$.
Step3: Simplify the exponent
Calculate $\frac{7}{2}-1=\frac{7 - 2}{2}=\frac{5}{2}$. So, $f^\prime(x)=\frac{7}{2}x^{\frac{5}{2}}$.
Answer:
$\frac{7}{2}x^{\frac{5}{2}}$