Given the function f(x)=(x+8)(3x+5-3x^(2)), find f^(')(x) in any form. Answer: f^(')(x)=

Write Function: Write down the function to be differentiated. f(x) = (x + 8)(3x + 5 - 3x^2)Apply Product Rule: Apply the product rule for differentiation, which 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 u = x + 8 and v = 3x + 5 - 3x^2. Then f(x) = uv. f'(x) = u'v + uv'Differentiate u: Differentiate u = x + 8 with respect to x. u' = d/dx (x + 8) = 1Differentiate v: Differentiate v = 3x + 5 - 3x^2 with respect to x. v' = d/dx (3x + 5 - 3x^2) = 3 - 6xSubstitute into Formula: Substitute u', u, v', and v into the product rule formula. f'(x) = u'v + uv' = (1)(3x + 5 - 3x^2) + (x + 8)(3 - 6x)Expand Terms: Expand the terms in the expression. f'(x) = (3x + 5 - 3x^2) + (3x + 24 - 6x^2 - 48x)Combine Like Terms: Combine like terms. f'(x) = 3x + 5 - 3x^2 + 3x + 24 - 6x^2 - 48x f'(x) = -9x^2 - 42x + 29

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Given the function
f(x)=(x+8)(3x+5-3x^(2)), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=(x+8)\left(3 x+5-3 x^{2}\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Given the function

f(x)=(x+8)\left(3 x+5-3 x^{2}\right)

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Write Function: Write down the function to be differentiated. $\newline$ $f(x) = (x + 8)(3x + 5 - 3x^2)$
Apply Product Rule: Apply the product rule for differentiation, which 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. $\newline$ Let $u = x + 8$ and $v = 3x + 5 - 3x^2$ . Then $f(x) = uv$ . $\newline$ $f'(x) = u'v + uv'$
Differentiate $u$ : Differentiate $u = x + 8$ with respect to $x$ . $\newline$ $u' = \frac{d}{dx} (x + 8) = 1$
Differentiate $v$ : Differentiate $v = 3x + 5 - 3x^2$ with respect to $x$ . $v' = \frac{d}{dx} (3x + 5 - 3x^2) = 3 - 6x$
Substitute into Formula: Substitute $u'$ , $u$ , $v'$ , and $v$ into the product rule formula. $\newline$ $f'(x) = u'v + uv' = (1)(3x + 5 - 3x^2) + (x + 8)(3 - 6x)$
Expand Terms: Expand the terms in the expression. $\newline$ $f'(x) = (3x + 5 - 3x^2) + (3x + 24 - 6x^2 - 48x)$
Combine Like Terms: Combine like terms. $\newline$ $f'(x) = 3x + 5 - 3x^2 + 3x + 24 - 6x^2 - 48x$ $\newline$ $f'(x) = -9x^2 - 42x + 29$