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Derivative Calculation for $f(x)=8(x^2 + 4x - 7)$
Mathematics
Grade 11 (Senior High School)
Question Content
Find the derivative of the function $f(x)=8(x^2 + 4x - 7)$
Correct Answer
$f'(x)=16x + 32$
Detailed Solution Steps
1
Step 1: Apply the constant multiple rule: $f'(x)=8\\cdot\\frac{d}{dx}[x^2 + 4x - 7]$.
2
Step 2: Differentiate the polynomial inside using the power rule and constant rule: $\\frac{d}{dx}[x^2 + 4x - 7]=2x + 4 - 0=2x + 4$.
3
Step 3: Multiply by the constant 8: $f'(x)=8\\cdot(2x + 4)=16x + 32$.
Knowledge Points Involved
1
Constant Multiple Rule of Differentiation
This rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function, expressed as $\\frac{d}{dx}[k\\cdot f(x)]=k\\cdot f'(x)$. It is used to simplify differentiation when a function is scaled by a constant factor.
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 the foundational rule for differentiating polynomial terms.
3
Sum/Difference Rule of Differentiation
This rule states that the derivative of a sum or difference of functions is the sum or difference of their derivatives, expressed as $\\frac{d}{dx}[f(x)\\pm g(x)]=f'(x)\\pm g'(x)$. It allows differentiating polynomial terms one by one.
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