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Given the function
y=-2(8x^(2)-9x-3)^(4), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=-2\left(8 x^{2}-9 x-3\right)^{4}$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Write a formula for an arithmetic sequence

Full solution

Q. Given the function

y=-2\left(8 x^{2}-9 x-3\right)^{4}

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Identify function type and rule: Identify the type of function and the rule needed to differentiate it. The function $y = -2(8x^2 - 9x - 3)^4$ is a composite function, where an inner function $(8x^2 - 9x - 3)$ is raised to a power and then multiplied by a constant. To differentiate this function, we will use the chain rule.
Apply chain rule: Apply the chain rule. 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. Let $u = 8x^2 - 9x - 3$ , so $y = -2u^4$ . The derivative of $y$ with respect to $u$ is $\frac{dy}{du} = -2 \times 4u^3 = -8u^3$ . Now we need to find $\frac{du}{dx}$ .
Find $\frac{du}{dx}$ : Differentiate the inner function $u = 8x^2 - 9x - 3$ with respect to $x$ . The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 16x - 9$ .
Substitute into chain rule: Substitute $\frac{du}{dx}$ into the chain rule formula. We have $\frac{dy}{du} = -8u^3$ and $\frac{du}{dx} = 16x - 9$ , so the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -8u^3 \cdot (16x - 9)$ .
Replace $u$ for final expression: Replace $u$ with the original inner function to get the final expression for $\frac{dy}{dx}$ . Substituting $u$ back into the expression, we get $\frac{dy}{dx} = -8(8x^2 - 9x - 3)^3 \cdot (16x - 9)$ .

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