Given the function y=(2x-1)^(4), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Apply Chain Rule: To find the derivative of the function y=(2x-1)^(4) with respect to x, we will use 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.Derivative of Outer Function: The outer function is u^4 and the inner function is u=2x-1. We first take the derivative of the outer function with respect to u, which is 4u^3.Derivative of Inner Function: Next, we take the derivative of the inner function with respect to x, which is the derivative of 2x-1. The derivative of 2x is 2, and the derivative of -1 is 0, so the derivative of the inner function is 2.Chain Rule Application: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us (dy)/(dx) = 4u^3 * 2.Substitute back u: Substitute u=2x-1 back into the expression to get the derivative in terms of x. This gives us (dy)/(dx) = 4(2x-1)^3 * 2.Simplify Expression: Finally, we simplify the expression by multiplying the constants. This gives us (dy)/(dx) = 8(2x-1)^3.

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Given the function
y=(2x-1)^(4), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=(2 x-1)^{4}$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 2

Csc, sec, and cot of special angles

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Q. Given the function

y=(2 x-1)^{4}

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Apply Chain Rule: To find the derivative of the function $y=(2x-1)^{4}$ with respect to $x$ , we will use 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.
Derivative of Outer Function: The outer function is $u^4$ and the inner function is $u=2x-1$ . We first take the derivative of the outer function with respect to $u$ , which is $4u^3$ .
Derivative of Inner Function: Next, we take the derivative of the inner function with respect to $x$ , which is the derivative of $2x-1$ . The derivative of $2x$ is $2$ , and the derivative of $-1$ is $0$ , so the derivative of the inner function is $2$ .
Chain Rule Application: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us $(\frac{dy}{dx}) = 4u^3 \times 2$ .
Substitute back $u$ : Substitute $u=2x-1$ back into the expression to get the derivative in terms of $x$ . This gives us $\frac{dy}{dx} = 4(2x-1)^3 \times 2$ .
Simplify Expression: Finally, we simplify the expression by multiplying the constants. This gives us $(\frac{dy}{dx} = 8(2x-1)^3)$ .