Given the function y=(10 x+10)(x^(3)-x^(2)+1), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Identify function: Identify the function to differentiate. We are given the function y=(10x+10)(x^3-x^2+1) and we need to find its derivative with respect to x, which is denoted as (dy)/(dx).Apply product rule: Apply the product rule for differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let's denote u=10x+10 and v=x^3-x^2+1. Then, (dy)/(dx) = u'(v) + u(v)'.Differentiate u: Differentiate u=10x+10 with respect to x. The derivative of u with respect to x is u' = d/dx(10x+10) = 10.Differentiate v: Differentiate v=x^3-x^2+1 with respect to x. The derivative of v with respect to x is v' = d/dx(x^3-x^2+1) = 3x^2 - 2x.Substitute into formula: Substitute u', v, and v' into the product rule formula. (dy)/(dx) = u'(v) + u(v)' = 10(x^3-x^2+1) + (10x+10)(3x^2-2x).Expand and simplify: Expand the terms and simplify the expression. (dy)/(dx) = 10x^3 - 10x^2 + 10 + (30x^3 - 20x^2 + 30x^2 - 20x).Combine like terms: Combine like terms. (dy)/(dx) = 10x^3 - 10x^2 + 10 + 30x^3 - 20x^2 + 30x^2 - 20x (dy)/(dx) = 40x^3 - 10x^2 + 30x^2 - 20x + 10 (dy)/(dx) = 40x^3 + 20x^2 - 20x + 10

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Given the function
y=(10 x+10)(x^(3)-x^(2)+1), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=(10 x+10)\left(x^{3}-x^{2}+1\right)$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 2

Evaluate exponential functions

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Q. Given the function

y=(10 x+10)\left(x^{3}-x^{2}+1\right)

, 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 $y=(10x+10)(x^3-x^2+1)$ and we need to find its derivative with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply product rule: Apply the product rule for differentiation. $\newline$ The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let's denote $u=10x+10$ and $v=x^3-x^2+1$ . Then, $\frac{dy}{dx} = u'(v) + u(v)'$ .
Differentiate $u$ : Differentiate $u=10x+10$ with respect to $x$ . The derivative of $u$ with respect to $x$ is $u' = \frac{d}{dx}(10x+10) = 10$ .
Differentiate $v$ : Differentiate $v=x^3-x^2+1$ with respect to $x$ . The derivative of $v$ with respect to $x$ is $v' = \frac{d}{dx}(x^3-x^2+1) = 3x^2 - 2x$ .
Substitute into formula: Substitute $u'$ , $v$ , and $v'$ into the product rule formula. $\newline$ $\frac{dy}{dx} = u'(v) + u(v)' = 10(x^3-x^2+1) + (10x+10)(3x^2-2x)$ .
Expand and simplify: Expand the terms and simplify the expression. $\newline$ $\frac{dy}{dx} = 10x^3 - 10x^2 + 10 + (30x^3 - 20x^2 + 30x^2 - 20x)$ .
Combine like terms: Combine like terms. $\newline$ $(dy)/(dx) = 10x^3 - 10x^2 + 10 + 30x^3 - 20x^2 + 30x^2 - 20x$ $\newline$ $(dy)/(dx) = 40x^3 - 10x^2 + 30x^2 - 20x + 10$ $\newline$ $(dy)/(dx) = 40x^3 + 20x^2 - 20x + 10$