(d)/(dx)(root(5)(x^(2)))=

Rewrite in Exponential Form: We need to find the derivative of the function $ f(x) = \sqrt[5]{x^2} $. To do this, we can rewrite the function in exponential form to make differentiation easier. The fifth root of $ x^2 $ can be written as $ x^{2/5} $.Apply Power Rule: Now, we apply the power rule for differentiation, which states that the derivative of $ x^n $ with respect to $ x $ is $ nx^{n-1} $. In this case, $ n = \frac{2}{5} $, so we get $ \frac{d}{dx}(x^{2/5}) = \frac{2}{5}x^{(2/5)-1} $.Simplify Exponent: Next, we simplify the exponent in the derivative. Subtracting 1 from $ \frac{2}{5} $ gives us $ \frac{2}{5} - \frac{5}{5} = \frac{-3}{5} $. So, the derivative is $ \frac{2}{5}x^{-3/5} $.Finalize Derivative: Finally, we can leave the answer in this form, or we can rewrite it to avoid negative exponents by placing $ x^{-3/5} $ in the denominator: $ \frac{2}{5x^{3/5}} $. This is the derivative of the fifth root of $ x^2 $ with respect to $ x $.

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Algebra 2

Simplify variable expressions using properties

Full solution

\frac{d}{d x}\left(\sqrt[5]{x^{2}}\right)=

Rewrite in Exponential Form: We need to find the derivative of the function $f(x) = \sqrt[5]{x^2}$ . To do this, we can rewrite the function in exponential form to make differentiation easier. The fifth root of $x^2$ can be written as $x^{2/5}$ .
Apply Power Rule: Now, we apply the power rule for differentiation, which states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$ . In this case, $n = \frac{2}{5}$ , so we get $\frac{d}{dx}(x^{2/5}) = \frac{2}{5}x^{(2/5)-1}$ .
Simplify Exponent: Next, we simplify the exponent in the derivative. Subtracting $1$ from $\frac{2}{5}$ gives us $\frac{2}{5} - \frac{5}{5} = \frac{-3}{5}$ . So, the derivative is $\frac{2}{5}x^{-3/5}$ .
Finalize Derivative: Finally, we can leave the answer in this form, or we can rewrite it to avoid negative exponents by placing $x^{-3/5}$ in the denominator: $\frac{2}{5x^{3/5}}$ . This is the derivative of the fifth root of $x^2$ with respect to $x$ .