Question

section 2.4: power and sum rule score: 40/180 answered: 4/18 question 5 find the derivative of: $sqrt4{x}$ hint: recall that $sqrtn{x}=x^{\frac{1}{n}}$. question help: video

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ using the rule $\sqrt[n]{x}=x^{\frac{1}{n}}$.

Step2: Apply the power - rule for derivatives

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $y=x^{\frac{1}{4}}$, we have $n=\frac{1}{4}$. So $y^\prime=\frac{1}{4}x^{\frac{1}{4}-1}$.

Step3: Simplify the exponent

Calculate $\frac{1}{4}-1=\frac{1 - 4}{4}=-\frac{3}{4}$. So $y^\prime=\frac{1}{4}x^{-\frac{3}{4}}$.

Answer:

$\frac{1}{4}x^{-\frac{3}{4}}$