(d)/(dx)(root(4)(x))=

Apply Power Rule: To find the derivative of the fourth root of x, which is x^(1/4), we will use the power rule for derivatives. The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1). In this case, n = 1/4.Differentiate x^(1/4): Applying the power rule, we differentiate x^(1/4) to get (1/4)*x^(1/4 - 1). We subtract 1 from the exponent 1/4 to get the new exponent for x.Simplify Exponent: Simplifying the new exponent, 1/4 - 1 equals -3/4. So the derivative is (1/4)*x^(-3/4).Final Derivative Form: The final simplified form of the derivative is (1/4)*x^(-3/4), which can also be written as (1/4)*(1/x^(3/4)) or (1/4)/(x^(3/4)).

Home

Math Problems

Algebra 2

Simplify variable expressions using properties

Full solution

\frac{d}{d x}(\sqrt[4]{x})=

Apply Power Rule: To find the derivative of the fourth root of $x$ , which is $x^{1/4}$ , we will use the power rule for derivatives. The power rule states that if $f(x) = x^n$ , then $f'(x) = n \cdot x^{n-1}$ . In this case, $n = \frac{1}{4}$ .
Differentiate $x^{\frac{1}{4}}$ : Applying the power rule, we differentiate $x^{\frac{1}{4}}$ to get $(\frac{1}{4})\cdot x^{\frac{1}{4} - 1}$ . We subtract $1$ from the exponent $\frac{1}{4}$ to get the new exponent for $x$ .
Simplify Exponent: Simplifying the new exponent, $\frac{1}{4} - 1$ equals $-\frac{3}{4}$ . So the derivative is $(\frac{1}{4})\cdot x^{-\frac{3}{4}}$ .
Final Derivative Form: The final simplified form of the derivative is $(\frac{1}{4})x^{(-\frac{3}{4})}$ , which can also be written as $(\frac{1}{4})(\frac{1}{x^{\frac{3}{4}}})$ or $(\frac{1}{4})/(x^{\frac{3}{4}})$ .