Given the function y=(4)/(5root(5)(x^(3))), find (dy)/(dx). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: (dy)/(dx)=

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Given the function  y=(4)/(5root(5)(x^(3))), find  (dy)/(dx). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer:  (dy)/(dx)=

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Rewrite function with fractional exponent: We are given the function y = (4)/(5√(5)(x^(3))). To find the derivative (dy)/(dx), we will use the power rule for differentiation, which states that if y = x^n, then (dy)/(dx) = n*x^(n-1). In this case, we need to rewrite the function in a form that allows us to apply the power rule.Combine constants and powers: First, we rewrite the square root in the denominator as a fractional exponent: √(5) = 5^(1/2). Then, we rewrite the function y as y = 4/(5*5^(1/2)*x^3). This can be further simplified by combining the constants and the powers of 5.Apply power rule for differentiation: The function becomes y = (4/5^(3/2)) * x^(-3). Now we can apply the power rule to find the derivative with respect to x.Simplify the derivative expression: Differentiating y with respect to x, we get (dy)/(dx) = (4/5^(3/2)) * (-3) * x^(-3 - 1). This simplifies to (dy)/(dx) = -12/(5^(3/2)) * x^(-4).Rewrite in radical form: We want to express the answer in radical form without using negative exponents. To do this, we rewrite x^(-4) as 1/x^4 and 5^(3/2) as √(5^3) or √(125).Finalize the derivative expression: The derivative (dy)/(dx) in radical form without negative exponents is (dy)/(dx) = -12/(√(125)*x^4). We can simplify √(125) to 5√(5).Finally, we simplify the fraction by dividing -12 by 5√(5). The derivative (dy)/(dx) is (dy)/(dx) = -12/(5√(5)*x^4).

Given the function $y=\frac{4}{5 \sqrt[5]{x^{3}}}$ , 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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Given the function $y=\frac{4}{5 \sqrt[5]{x^{3}}}$ , find $\frac{d y}{d x}$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $\frac{d y}{d x}=$