How to Differentiate $f(x)=2(x^4 + x)$ - Math Solver AI

$Math Problem$

Mathematics Grade 11 (Senior High School)

Question Content

Find the derivative of the function $f(x)=2(x^4 + x)$

Correct Answer

$f'(x)=8x^3 + 2$

Detailed Solution Steps

1
Step 1: Use the constant multiple rule: $f'(x)=2\\cdot\\frac{d}{dx}[x^4 + x]$.
2
Step 2: Differentiate $x^4 + x$ with the power rule: $\\frac{d}{dx}[x^4 + x]=4x^3 + 1x^{0}=4x^3 + 1$.
3
Step 3: Multiply by the constant 2: $f'(x)=2\\cdot(4x^3 + 1)=8x^3 + 2$.

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, including the term $x$ where $n=1$.

3

Sum Rule of Differentiation

This rule states that the derivative of a sum of functions is the sum of their derivatives, expressed as $\\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)$. It enables term-by-term differentiation of polynomial sums.

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