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

Question

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

Bytelearn · Accepted Answer

Identify function: Identify the function to differentiate. We are given the function y = (4)/(root(6)(8x^2 - 6x + 9)). We need to find the derivative of this function with respect to x, which is denoted as (dy)/(dx).Rewrite with exponents: Rewrite the function using exponents. The sixth root can be written as a fractional exponent. So, we rewrite the function as y = 4 * (8x^2 - 6x + 9)^(-1/6).Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is u^(-1/6) and the inner function is (8x^2 - 6x + 9).Differentiate outer function: Differentiate the outer function with respect to the inner function. The derivative of u^(-1/6) with respect to u is (-1/6) * u^(-7/6). We will substitute u back with (8x^2 - 6x + 9) later.Differentiate inner function: Differentiate the inner function with respect to x. The derivative of (8x^2 - 6x + 9) with respect to x is 16x - 6.Combine using chain rule: Combine the derivatives using the chain rule. (dy)/(dx) = (-1/6) * (8x^2 - 6x + 9)^(-7/6) * (16x - 6).Simplify expression: Simplify the expression. We can multiply the constants together and write the derivative in a simplified form. (dy)/(dx) = (-1/6) * 4 * (16x - 6) * (8x^2 - 6x + 9)^(-7/6) (dy)/(dx) = (-4/6) * (16x - 6) * (8x^2 - 6x + 9)^(-7/6) (dy)/(dx) = (-2/3) * (16x - 6) * (8x^2 - 6x + 9)^(-7/6)

Given the function $\newline$ $y=\frac{4}{\sqrt[6]{8x^{2}-6x+9}},$ find $\newline$ $\frac{dy}{dx}$ in any form.

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Given the function $\newline$ $y=\frac{4}{\sqrt[6]{8x^{2}-6x+9}},$ find $\newline$ $\frac{dy}{dx}$ in any form.