Given the function f(x)=(7x^(2)+6x)^(4), find f^(')(x) in any form. Answer: f^(')(x)=

Identify Functions: We need to find the derivative of the function f(x)=(7x^(2)+6x)^(4). To do this, we will use the chain rule, which 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.Find Outer Derivative: First, let's identify the outer function and the inner function. The outer function is g(u)=u^(4) and the inner function is h(x)=7x^(2)+6x. We will find the derivative of each function separately.Find Inner Derivative: The derivative of the outer function g(u)=u^(4) with respect to u is g'(u)=4u^(3).Apply Chain Rule: The derivative of the inner function h(x)=7x^(2)+6x with respect to x is h'(x)=14x+6.Substitute Derivatives: Now, we apply the chain rule. The derivative of the composite function f(x) is f'(x)=g'(h(x)) * h'(x). We substitute the derivatives we found into this formula.Leave in Factored Form: Substituting the derivatives, we get f'(x)=4(7x^(2)+6x)^(3) * (14x+6).We can leave the derivative in this factored form, or we can expand it if required. However, the problem does not specify that we need to expand, so we will leave it in the factored form.

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Given the function
f(x)=(7x^(2)+6x)^(4), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=\left(7 x^{2}+6 x\right)^{4}$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Math Problems

Algebra 1

Multiplication with rational exponents

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Q. Given the function

f(x)=\left(7 x^{2}+6 x\right)^{4}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: We need to find the derivative of the function $f(x)=(7x^{2}+6x)^{4}$ . To do this, we will use the chain rule, which 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.
Find Outer Derivative: First, let's identify the outer function and the inner function. The outer function is $g(u)=u^{4}$ and the inner function is $h(x)=7x^{2}+6x$ . We will find the derivative of each function separately.
Find Inner Derivative: The derivative of the outer function $g(u)=u^{4}$ with respect to $u$ is $g'(u)=4u^{3}$ .
Apply Chain Rule: The derivative of the inner function $h(x)=7x^{2}+6x$ with respect to $x$ is $h'(x)=14x+6$ .
Substitute Derivatives: Now, we apply the chain rule. The derivative of the composite function $f(x)$ is $f'(x)=g'(h(x)) \cdot h'(x)$ . We substitute the derivatives we found into this formula.
Leave in Factored Form: Substituting the derivatives, we get $f'(x)=4(7x^{2}+6x)^{3} \cdot (14x+6)$ .
Leave in Factored Form: Substituting the derivatives, we get $f'(x)=4(7x^{2}+6x)^{3} \times (14x+6)$ .We can leave the derivative in this factored form, or we can expand it if required. However, the problem does not specify that we need to expand, so we will leave it in the factored form.