Question

(d) find $d_x(x^{\frac{1}{4}})$. $d_x(x^{\frac{1}{4}})=square$

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=\frac{d}{dx}(x^n)=nx^{n - 1}$. Here $n=\frac{1}{4}$.

Step2: Apply the power - rule

Let $y = x^{\frac{1}{4}}$. Then $D_x(x^{\frac{1}{4}})=\frac{1}{4}x^{\frac{1}{4}-1}$.

Step3: Simplify the exponent

$\frac{1}{4}-1=\frac{1 - 4}{4}=-\frac{3}{4}$. So $D_x(x^{\frac{1}{4}})=\frac{1}{4}x^{-\frac{3}{4}}$.

Answer:

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