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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=3x3 y=3 \sqrt{x^{3}} , find dydx \frac{d y}{d x} . Express your answer in radical form without using negative exponents, simplifying all fractions.\newlineAnswer: dydx= \frac{d y}{d x}=

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Q. Given the function y=3x3 y=3 \sqrt{x^{3}} , find dydx \frac{d y}{d x} . Express your answer in radical form without using negative exponents, simplifying all fractions.\newlineAnswer: dydx= \frac{d y}{d x}=
  1. Identify Function: Identify the function to differentiate.\newlineWe are given the function y=3x3y = 3\sqrt{x^3}, which can be rewritten as y=3(x32)y = 3(x^{\frac{3}{2}}) to make differentiation easier.
  2. Apply Power Rule: Apply the power rule for differentiation.\newlineThe power rule states that the derivative of xnx^n with respect to xx is n∗x(n−1)n*x^{(n-1)}. Using this rule, we differentiate y=3(x32)y = 3(x^{\frac{3}{2}}).\newlinedydx=3⋅(32)⋅x(32−1)\frac{dy}{dx} = 3 \cdot \left(\frac{3}{2}\right) \cdot x^{\left(\frac{3}{2}-1\right)}
  3. Simplify Expression: Simplify the expression.\newline(dydx)=92⋅x12(\frac{dy}{dx}) = \frac{9}{2} \cdot x^{\frac{1}{2}}\newlineSince x12x^{\frac{1}{2}} is the same as x\sqrt{x}, we can write the derivative as:\newline(dydx)=92x(\frac{dy}{dx}) = \frac{9}{2}\sqrt{x}

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