Differentiate the function y = 3√x2 (2x – x2)

Q: Differentiate the function y = 3√x2 (2x – x2)

Differentiate the function y = 3√x2 (2x - x2) dy/dx = ($10x^{2/3}$)/3 - ($8x^{5/3}$)/3 ✓ dy/dx = ($10x^{5/3}$)/3 - ($8x^{2/3}$)/3 dy/dx = ($10x^{2/3}$)/3 - ($8x^{2/3}$)/3 dy/dx = ($10x^{5/3}$)/3 - ($8x^{5/3}$)/3 Explanation First, rewrite using exponents: cube root of x² equals x to the power two-thirds. Expand the product: x to the power two-thirds times parenthesis 2x minus x² equals 2x to the power five-thirds minus x to the power eight-thirds. Differentiate each term using the power rule. For 2x to the power five-thirds, bring down the exponent: 2 × five-thirds × x to the power two-thirds equals ten-thirds × x to the power two-thirds. For negative x to the power eight-thirds, bring down the exponent: negative 1 × eight-thirds × x to the power five-thirds equals negative eight-thirds × x to the power five-thirds. Combine: dy over dx equals ten-thirds x to the power two-thirds minus eight-thirds x to the power five-thirds.

Question

Answer 1

Differentiate the function y = 3√x2 (2x – x2)

dy/dx = ($10x^{2/3}$)/3 – ($8x^{5/3}$)/3 ✓
dy/dx = ($10x^{5/3}$)/3 – ($8x^{2/3}$)/3
dy/dx = ($10x^{2/3}$)/3 – ($8x^{2/3}$)/3
dy/dx = ($10x^{5/3}$)/3 – ($8x^{5/3}$)/3

Explanation

First, rewrite using exponents: cube root of x² equals x to the power two-thirds. Expand the product: x to the power two-thirds times parenthesis 2x minus x² equals 2x to the power five-thirds minus x to the power eight-thirds.

Differentiate each term using the power rule. For 2x to the power five-thirds, bring down the exponent: 2 × five-thirds × x to the power two-thirds equals ten-thirds × x to the power two-thirds.

For negative x to the power eight-thirds, bring down the exponent: negative 1 × eight-thirds × x to the power five-thirds equals negative eight-thirds × x to the power five-thirds. Combine: dy over dx equals ten-thirds x to the power two-thirds minus eight-thirds x to the power five-thirds.