Let h(x)=root(3)(x^(2)). h^(')(x)=

Identify Function: We are given the function h(x) = ∛(x²). To find the derivative h'(x), we will use the chain rule. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x). In this case, our outer function f(u) is ∛(u) and our inner function g(x) is x².Derivative of Outer Function: First, we need to find the derivative of the outer function f(u) = ∛(u) with respect to u. We can rewrite this function as u^(1/3), and using the power rule, we find that the derivative f'(u) = (1/3)u^(-2/3).Derivative of Inner Function: Next, we find the derivative of the inner function g(x) = x². Using the power rule, the derivative g'(x) = 2x.Apply Chain Rule: Now we apply the chain rule. We multiply the derivative of the outer function by the derivative of the inner function. This gives us h'(x) = f'(g(x)) * g'(x) = (1/3)(x²)^(-2/3) * 2x.Combine Terms: We simplify the expression by combining the terms. The x in 2x cancels out one of the x's in (x²)^(-2/3), leaving us with h'(x) = (1/3) * 2 * x^(-1/3) = (2/3)x^(-1/3).Simplify Expression: Finally, we can rewrite the expression with a positive exponent by moving x^(-1/3) to the denominator, which gives us h'(x) = (2/3) / x^(1/3).

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Precalculus

Find trigonometric ratios using multiple identities

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Q. Let

h(x)=\sqrt[3]{x^{2}}

\newline

h^{\prime}(x)=

Identify Function: We are given the function $h(x) = \sqrt[3]{x^2}$ . To find the derivative $h'(x)$ , we will use the chain rule. The chain rule states that if you have a composite function $f(g(x))$ , the derivative is $f'(g(x)) \cdot g'(x)$ . In this case, our outer function $f(u)$ is $\sqrt[3]{u}$ and our inner function $g(x)$ is $x^2$ .
Derivative of Outer Function: First, we need to find the derivative of the outer function $f(u) = \sqrt[3]{u}$ with respect to $u$ . We can rewrite this function as $u^{\frac{1}{3}}$ , and using the power rule, we find that the derivative $f'(u) = \frac{1}{3}u^{-\frac{2}{3}}$ .
Derivative of Inner Function: Next, we find the derivative of the inner function $g(x) = x^2$ . Using the power rule, the derivative $g'(x) = 2x$ .
Apply Chain Rule: Now we apply the chain rule. We multiply the derivative of the outer function by the derivative of the inner function. This gives us $h'(x) = f'(g(x)) \cdot g'(x) = \left(\frac{1}{3}\right)(x^2)^{-\frac{2}{3}} \cdot 2x$ .
Combine Terms: We simplify the expression by combining the terms. The $x$ in $2x$ cancels out one of the $x$ 's in $(x^2)^{-\frac{2}{3}}$ , leaving us with $h'(x) = \frac{1}{3} \times 2 \times x^{-\frac{1}{3}} = \frac{2}{3}x^{-\frac{1}{3}}$ .
Simplify Expression: Finally, we can rewrite the expression with a positive exponent by moving $x^{-1/3}$ to the denominator, which gives us $h'(x) = \frac{2}{3} / x^{1/3}$ .