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Derivative of $f(x)=(x^2 + 3)^4$ Using the Chain Rule
Mathematics
Grade 12 (Senior High School)
Question Content
Find the derivative of the function $f(x)=(x^2 + 3)^4$
Correct Answer
$f'(x)=8x(x^2 + 3)^3$
Detailed Solution Steps
1
Step 1: Recognize this is a composite function, so apply the chain rule: If $f(x)=h(g(x))$, then $f'(x)=h'(g(x))\\cdot g'(x)$. Let $u=g(x)=x^2 + 3$ and $h(u)=u^4$.
2
Step 2: Differentiate $h(u)=u^4$ using the power rule: $h'(u)=4u^3$. Substitute back $u=x^2 + 3$: $h'(g(x))=4(x^2 + 3)^3$.
3
Step 3: Differentiate $g(x)=x^2 + 3$ using the power rule: $g'(x)=2x + 0=2x$.
4
Step 4: Multiply the two results from Step 2 and Step 3: $f'(x)=4(x^2 + 3)^3\\cdot2x=8x(x^2 + 3)^3$.
Knowledge Points Involved
1
Chain Rule of Differentiation
The chain rule is used to differentiate composite functions, expressed as $\\frac{d}{dx}[h(g(x))]=h'(g(x))\\cdot g'(x)$, where $h(u)$ is the outer function and $u=g(x)$ is the inner function. It applies when one function is nested inside another.
2
Power Rule of Differentiation
The power rule defines the derivative of a monomial $x^n$ (where $n$ is a real number) as $\\frac{d}{dx}[x^n]=nx^{n-1}$. It is used here to differentiate both the outer power function and the inner polynomial function.
3
Constant Rule of Differentiation
This rule states that the derivative of a constant value $c$ is 0, since a constant function has a slope of 0 at all points, expressed as $\\frac{d}{dx}[c]=0$. It is used to differentiate the constant term 3 in the inner function.
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