Given the function y=4(9x^(2)-8x)^((6)/(5)) find (dy)/(dx) in any form.

Question

Given the function  y=4(9x^(2)-8x)^((6)/(5)) find  (dy)/(dx) in any form.

Bytelearn · Accepted Answer

Identify Functions: We need to find the derivative of the function y with respect to x. The function is y=4(9x^(2)-8x)^((6)/(5)). We will use the chain rule to differentiate this function. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Apply Chain Rule: First, let's identify the outer function and the inner function. The outer function is u^(6/5) where u is the inner function, and the inner function is u=9x^2-8x. We will differentiate the outer function with respect to u and then multiply by the derivative of the inner function with respect to x.Differentiate Outer Function: Differentiate the outer function with respect to u. The derivative of u^(6/5) with respect to u is (6/5)u^((6/5)-1) = (6/5)u^(1/5).Differentiate Inner Function: Now, differentiate the inner function u=9x^2-8x with respect to x. The derivative of 9x^2 with respect to x is 18x, and the derivative of -8x with respect to x is -8. So, the derivative of u with respect to x is du/dx = 18x - 8.Apply Chain Rule: Now we apply the chain rule. Multiply the derivative of the outer function by the derivative of the inner function. This gives us dy/dx = (6/5)(9x^2-8x)^(1/5) * (18x - 8).Final Derivative Calculation: Finally, we multiply the constant 4 from the original function by the derivative we found in the previous step. This gives us dy/dx = 4 * (6/5)(9x^2-8x)^(1/5) * (18x - 8).

Given the function $\newline$ $y=4(9x^{2}-8x)^{\frac{6}{5}}$ find $\frac{dy}{dx}$ in any form.

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Given the function $\newline$ $y=4(9x^{2}-8x)^{\frac{6}{5}}$ find $\frac{dy}{dx}$ in any form.