How to Differentiate a Polynomial and Find its Tangent

Differentiates a polynomial and gives the slope and the tangent line at a point you choose. Coefficients run from the highest power down, so 1, -3, 2 means x² − 3x + 2. The derivative of xⁿ is nxⁿ⁻¹, one degree lower.

This differentiates a polynomial and gives the slope and the tangent line at a point of your choosing.

\left(x^n\right)' = nx^{n-1}

For each term, bring the exponent down in front and reduce it by one. Constant terms vanish. Enter the coefficients from the highest power down, so $1, -3, 2$ means $x^2 - 3x + 2$ .

The tangent

Putting a value of $x$ into the derivative gives the slope of the graph there. The tangent is the line through $(x_0, f(x_0))$ with that slope.

y = f'(x_0)(x - x_0) + f(x_0)

Example

Differentiating $f(x) = x^2 - 3x + 2$ gives $f'(x) = 2x - 3$ . At $x = 2$ , $f(2) = 4 - 6 + 2 = 0$ and $f'(2) = 4 - 3 = 1$ . The tangent is $y = 1 \times (x - 2) + 0$ , that is $y = x - 2$ .

What the derivative tells you

The derivative says how steeply the graph is climbing at a point. Positive means rising, negative means falling, and 0 means a peak, a trough or a flat stretch. Solving $f'(x) = 0$ turns up the candidates for a maximum or a minimum.

Notes

Include a 0 for any missing power. For $x^3 + 1$ , enter $1, 0, 0, 1$ .

Only polynomials are handled here. Trigonometric and exponential functions cannot be entered.