Given the function y=(-5x-9)^(4), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Identify Functions: To find the derivative of the function y=(-5x-9)^(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: First, let's identify the outer function and the inner function. The outer function is u^4 and the inner function is u = -5x - 9. We will take the derivative of the outer function with respect to the inner function, which is 4u^3.Derivative of Inner Function: Now we will take the derivative of the inner function, u = -5x - 9, with respect to x. The derivative of -5x is -5 and the derivative of a constant (-9) is 0. So, the derivative of the inner function is -5.Apply Chain Rule: Next, we will apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us (4u^3)(-5), where u = -5x - 9.Substitute Back: Substitute u back into the expression to get the derivative in terms of x. This gives us (4(-5x - 9)^3)(-5).Simplify Expression: Finally, we simplify the expression to get the derivative of y with respect to x. This results in -20(-5x - 9)^3.

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Given the function
y=(-5x-9)^(4), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=(-5 x-9)^{4}$ , 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=(-5 x-9)^{4}

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Identify Functions: To find the derivative of the function $y=(-5x-9)^{4}$ 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: First, let's identify the outer function and the inner function. The outer function is $u^4$ and the inner function is $u = -5x - 9$ . We will take the derivative of the outer function with respect to the inner function, which is $4u^3$ .
Derivative of Inner Function: Now we will take the derivative of the inner function, $u = -5x - 9$ , with respect to $x$ . The derivative of $-5x$ is $-5$ and the derivative of a constant $(-9)$ is $0$ . So, the derivative of the inner function is $-5$ .
Apply Chain Rule: Next, we will apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us $(4u^3)(-5)$ , where $u = -5x - 9$ .
Substitute Back: Substitute $u$ back into the expression to get the derivative in terms of $x$ . This gives us $(4(-5x - 9)^3)(-5)$ .
Simplify Expression: Finally, we simplify the expression to get the derivative of $y$ with respect to $x$ . This results in $-20(-5x - 9)^3$ .