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$Given the function y=3sqrt(x^(3)), find (dy)/(dx). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: (dy)/(dx)=$

Given the function $y=3 \sqrt{x^{3}}$ , find $\frac{d y}{d x}$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $\frac{d y}{d x}=$

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Find trigonometric ratios of special angles

Full solution

Q. Given the function

y=3 \sqrt{x^{3}}

, find

\frac{d y}{d x}

. Express your answer in radical form without using negative exponents, simplifying all fractions.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = 3\sqrt{x^3}$ , which can be rewritten as $y = 3(x^{\frac{3}{2}})$ to make differentiation easier.
Apply Power Rule: Apply the power rule for differentiation. $\newline$ The power rule states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . Using this rule, we differentiate $y = 3(x^{\frac{3}{2}})$ . $\newline$ $\frac{dy}{dx} = 3 \cdot \left(\frac{3}{2}\right) \cdot x^{\left(\frac{3}{2}-1\right)}$
Simplify Expression: Simplify the expression. $\newline$ $(\frac{dy}{dx}) = \frac{9}{2} \cdot x^{\frac{1}{2}}$ $\newline$ Since $x^{\frac{1}{2}}$ is the same as $\sqrt{x}$ , we can write the derivative as: $\newline$ $(\frac{dy}{dx}) = \frac{9}{2}\sqrt{x}$

More problems from Find trigonometric ratios of special angles

Question

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Question

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Question

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