Question

hw10 differentiation rules 1 (targets c1, c5; §3.3)
score: 3/9 answered: 3/9
question 4
find the derivative of: $sqrt4{x}$
hint: recall that $sqrtn{x}=x^{\frac{1}{n}}$
question help: video message instructor
submit question

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 power - rule for differentiation

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

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

Step4: Rewrite with positive exponent

$x^{-\frac{3}{4}}=\frac{1}{x^{\frac{3}{4}}}$, so $y^\prime=\frac{1}{4x^{\frac{3}{4}}}=\frac{1}{4\sqrt[4]{x^{3}}}$.

Answer:

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