Given the function y=3(6x+5)^(5), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Identify Function: Identify the function to differentiate. We are given the function y = 3(6x+5)^(5) and we need to find its derivative with respect to x, which is denoted as (dy)/(dx).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 evaluated at the inner function times the derivative of the inner function. In this case, the outer function is u^5 and the inner function is u = 6x+5. Let's differentiate the outer function first, keeping the inner function as is. (d/dx)[3(u)^5] = 3 * 5 * u^(5-1) = 15u^4Differentiate Inner Function: Differentiate the inner function. Now we differentiate the inner function u = 6x+5 with respect to x. (d/dx)[6x+5] = 6Multiply Derivatives: Apply the chain rule by multiplying the derivatives. Now we multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative. (dy)/(dx) = 15u^4 * 6Substitute Inner Function: Substitute back the inner function. Replace u with the original inner function (6x+5). (dy)/(dx) = 15(6x+5)^4 * 6Simplify Expression: Simplify the expression. Now we simplify the expression by multiplying the constants. (dy)/(dx) = 90(6x+5)^4

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Given the function
y=3(6x+5)^(5), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=3(6 x+5)^{5}$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Q. Given the function

y=3(6 x+5)^{5}

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = 3(6x+5)^{5}$ and we need to find its derivative with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ 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. In this case, the outer function is $u^5$ and the inner function is $u = 6x+5$ . $\newline$ Let's differentiate the outer function first, keeping the inner function as is. $\newline$ $\frac{d}{dx}[3(u)^5]$ = $3 \times 5 \times u^{5-1}$ = $15u^4$
Differentiate Inner Function: Differentiate the inner function. $\newline$ Now we differentiate the inner function $u = 6x+5$ with respect to $x$ . $\newline$ $\frac{d}{dx}[6x+5] = 6$
Multiply Derivatives: Apply the chain rule by multiplying the derivatives. $\newline$ Now we multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative. $\newline$ $\frac{dy}{dx} = 15u^4 \times 6$
Substitute Inner Function: Substitute back the inner function. $\newline$ Replace $u$ with the original inner function $(6x+5)$ . $\newline$ $\frac{dy}{dx} = 15(6x+5)^4 \times 6$
Simplify Expression: Simplify the expression. $\newline$ Now we simplify the expression by multiplying the constants. $\newline$ $(\frac{dy}{dx} = 90(6x+5)^4)$