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Given the function
f(x)=(7x^(2)+2x+5)^(6), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=\left(7 x^{2}+2 x+5\right)^{6}$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Find the vertex of the transformed function

Full solution

Q. Given the function

f(x)=\left(7 x^{2}+2 x+5\right)^{6}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: We need to find the derivative of the function $f(x)=(7x^{2}+2x+5)^{6}$ . 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 Derivatives: First, let's identify the outer function and the inner function. The outer function is $g(u)=u^{6}$ and the inner function is $u(x)=7x^{2}+2x+5$ . We will find the derivative of the outer function with respect to $u$ , $g'(u)$ , and the derivative of the inner function with respect to $x$ , $u'(x)$ .
Apply Chain Rule: The derivative of the outer function $g(u)=u^{6}$ with respect to $u$ is $g'(u)=6u^{5}$ . We will substitute $u$ with the inner function later.
Calculate Derivative: The derivative of the inner function $u(x)=7x^{2}+2x+5$ with respect to $x$ is $u'(x)=14x+2$ , using the power rule and the fact that the derivative of a constant is zero.
Finalize Solution: Now we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is $f'(x)=g'(u) \cdot u'(x)$ , where $u$ is the inner function. Substituting the derivatives we found, we get $f'(x)=6(7x^{2}+2x+5)^{5} \cdot (14x+2)$ .
Finalize Solution: Now we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is $f'(x)=g'(u) \cdot u'(x)$ , where $u$ is the inner function. Substituting the derivatives we found, we get $f'(x)=6(7x^{2}+2x+5)^{5} \cdot (14x+2)$ .We can leave the derivative in this factored form, or we can expand it if necessary. However, the question does not require us to expand, so we will leave it in the factored form.

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