Calculus • Step-by-Step

Derivative of x⁴ - 4x³ + 6x² - 4x + 1

Differentiate a 4th-degree quartic polynomial step by step using term-by-term power rule to get 4x³ - 12x² + 12x - 4.

Problem Statement

\frac{d}{dx}[x^4 - 4x^3 + 6x^2 - 4x + 1]

Final Result

f'(x) = 4 \cdot x^{3}-12 \cdot x^{2}+12 \cdot x-4

Formula Required: Polynomial Term-by-Term Differentiation

Apply the power rule to each power of x independently.

\frac{d}{dx}[\sum c_i x^i] = \sum i c_i x^{i-1}

8 Steps

Set up the differentiation problem with respect to variable $x$.

\frac{d}{dx} \left[ x^{4}-4 \cdot x^{3}+6 \cdot x^{2}-4 \cdot x+1 \right]

Apply the Sum & Difference Rule: differentiate each term separately.

\frac{d}{dx} \left[ x^{4}-4 \cdot x^{3}+6 \cdot x^{2}-4 \cdot x+1 \right] = \frac{d}{dx}\left[ x^{4} \right] - \frac{d}{dx}\left[ -4 \cdot x^{3} \right] + \frac{d}{dx}\left[ 6 \cdot x^{2} \right] - \frac{d}{dx}\left[ -4 \cdot x \right] + \frac{d}{dx}\left[ 1 \right]

Differentiate $x^{4}$ using the Power Rule.

\frac{d}{dx}\left[ x^{4} \right] = 4 \cdot x^{3}

Differentiate $4 \cdot x^{3}$ using the Power Rule.

\frac{d}{dx}\left[ 4 \cdot x^{3} \right] = 12 \cdot x^{2}

Differentiate $6 \cdot x^{2}$ using the Power Rule.

\frac{d}{dx}\left[ 6 \cdot x^{2} \right] = 12 \cdot x

The derivative of linear term $4 \cdot x$ is its coefficient.

\frac{d}{dx}\left[ 4 \cdot x \right] = 4

The derivative of constant $1$ is $0$.

\frac{d}{dx}\left[ 1 \right] = 0

Combine all term derivatives and simplify algebraic terms.

\frac{d}{dx}\left[ x^{4}-4 \cdot x^{3}+6 \cdot x^{2}-4 \cdot x+1 \right] = 4 \cdot x^{3}-12 \cdot x^{2}+12 \cdot x-4

Interactive Calculator

Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.

Differentiate each term individually: d/dx[x⁴] = 4x³, d/dx[-4x³] = -12x², d/dx[6x²] = 12x, d/dx[-4x] = -4, and d/dx[1] = 0.