Given the function y=root(3)(27 x)sin x, find (dy)/(dx) in any form.

Identify Function Components: Identify the function components to differentiate. The function y = root(3)(27x)sin(x) can be rewritten as y = (27x)^(1/3) * sin(x). This function is a product of two functions: u(x) = (27x)^(1/3) and v(x) = sin(x).Apply Product Rule: Apply the product rule for differentiation. The product rule states that the derivative of a product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x). We will use this rule to find (dy)/(dx).Differentiate First Function: Differentiate the first function u(x) = (27x)^(1/3). Using the power rule, the derivative of u with respect to x is u'(x) = (1/3)(27x)^(-2/3) * 27 = 9(27x)^(-2/3).Differentiate Second Function: Differentiate the second function v(x) = sin(x). The derivative of v with respect to x is v'(x) = cos(x).Apply Product Rule: Apply the product rule using the derivatives from steps 3 and 4. (dy)/(dx) = u'(x)v(x) + u(x)v'(x) = 9(27x)^(-2/3) * sin(x) + (27x)^(1/3) * cos(x).Simplify Expression: Simplify the expression if possible. The expression is already in a simplified form, so we can state the final answer. (dy)/(dx) = 9(27x)^(-2/3) * sin(x) + (27x)^(1/3) * cos(x).

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Given the function $y=\sqrt[3]{27x}\sin x$ , find $\frac{dy}{dx}$ in any form.

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Algebra 1

Domain and range of square root functions: equations

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Q. Given the function

y=\sqrt[3]{27x}\sin x

, find

\frac{dy}{dx}

in any form.

Identify Function Components: Identify the function components to differentiate. $\newline$ The function $y = \sqrt[3]{27x}\sin(x)$ can be rewritten as $y = (27x)^{\frac{1}{3}} \cdot \sin(x)$ . This function is a product of two functions: $u(x) = (27x)^{\frac{1}{3}}$ and $v(x) = \sin(x)$ .
Apply Product Rule: Apply the product rule for differentiation. $\newline$ The product rule states that the derivative of a product of two functions $u(x)$ and $v(x)$ is given by $u'(x)v(x) + u(x)v'(x)$ . We will use this rule to find $\frac{dy}{dx}$ .
Differentiate First Function: Differentiate the first function $u(x) = (27x)^{\frac{1}{3}}$ . Using the power rule, the derivative of $u$ with respect to $x$ is $u'(x) = \frac{1}{3}(27x)^{-\frac{2}{3}} \times 27 = 9(27x)^{-\frac{2}{3}}$ .
Differentiate Second Function: Differentiate the second function $v(x) = \sin(x)$ . $\newline$ The derivative of $v$ with respect to $x$ is $v'(x) = \cos(x)$ .
Apply Product Rule: Apply the product rule using the derivatives from steps $3$ and $4$ . $\newline$ $(\frac{dy}{dx}) = u'(x)v(x) + u(x)v'(x) = 9(27x)^{-\frac{2}{3}} \cdot \sin(x) + (27x)^{\frac{1}{3}} \cdot \cos(x)$ .
Simplify Expression: Simplify the expression if possible. $\newline$ The expression is already in a simplified form, so we can state the final answer. $\newline$ $\frac{dy}{dx} = 9(27x)^{-\frac{2}{3}} \cdot \sin(x) + (27x)^{\frac{1}{3}} \cdot \cos(x)$ .