Differentiating Polynomials

Key Concepts

Differentiation finds the rate of change of a function — geometrically, it gives the gradient of the curve at any point.

The power rule

For any real number $n$:

$\frac{d}{dx}(x^n) = nx^{n-1}$

This works for positive integers, negative powers, and fractions.

Constants and sums

• The derivative of a constant $c$ is $0$
• Constants multiply through: $\frac{d}{dx}(cf(x)) = c \cdot f'(x)$
• Differentiate term by term: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

Worked Examples

Example 1: Basic power rule

Differentiate $y = x^5$

$\frac{dy}{dx} = 5x^4$

Example 2: Polynomial with multiple terms

Differentiate $f(x) = 3x^4 - 2x^3 + 7x - 5$

$f'(x) = 12x^3 - 6x^2 + 7$

Example 3: Rewriting before differentiating

Differentiate $y = \frac{4}{x^2} + 3\sqrt{x}$

Rewrite: $y = 4x^{-2} + 3x^{1/2}$

$\frac{dy}{dx} = -8x^{-3} + \frac{3}{2}x^{-1/2} = -\frac{8}{x^3} + \frac{3}{2\sqrt{x}}$

Example 4: Expanding first

Differentiate $y = x^2(3x - 1)$

Expand: $y = 3x^3 - x^2$

$\frac{dy}{dx} = 9x^2 - 2x$

Example 5: Finding the gradient at a point

Find the gradient of $y = 2x^3 - x + 4$ at the point where $x = 2$

$\frac{dy}{dx} = 6x^2 - 1$

At $x = 2$: $\frac{dy}{dx} = 6(4) - 1 = 23$