Question

2.4 homework ~ derivatives of constants, sums and powers
score: 40/250 answered: 4/25
question 5
find the derivative of: $sqrt4{x}$
hint: recall that $sqrtn{x} = x^{\frac{1}{n}}$

Explanation:

Step1: Rewrite the radical as a power

The fourth root of \( x \), \( \sqrt[4]{x} \), can be written as \( x^{\frac{1}{4}} \) using the rule \( \sqrt[n]{x}=x^{\frac{1}{n}} \).

Step2: Apply the power rule for differentiation

The power rule states that if \( f(x) = x^n \), then \( f'(x)=nx^{n - 1} \). Here, \( n=\frac{1}{4} \), so we differentiate \( x^{\frac{1}{4}} \) with respect to \( x \).
Using the power rule, we get \( \frac{1}{4}x^{\frac{1}{4}-1} \).

Step3: Simplify the exponent

Simplify \( \frac{1}{4}-1=\frac{1}{4}-\frac{4}{4}=-\frac{3}{4} \). So the derivative is \( \frac{1}{4}x^{-\frac{3}{4}} \).

Step4: Rewrite the negative exponent as a radical (optional, but common)

We can rewrite \( x^{-\frac{3}{4}} \) as \( \frac{1}{x^{\frac{3}{4}}} \) and \( x^{\frac{3}{4}}=\sqrt[4]{x^3} \), so the derivative can also be written as \( \frac{1}{4\sqrt[4]{x^3}} \). But both forms are correct, and \( \frac{1}{4}x^{-\frac{3}{4}} \) is also acceptable.

Answer:

\( \frac{1}{4}x^{-\frac{3}{4}} \) (or \( \frac{1}{4\sqrt[4]{x^3}} \))