Derivative — with a detailed breakdown of every step and a graph
Function f(x)
Derivative
How it was calculated — step by step
Every step names the rule that was applied and shows what was substituted, so the solution can be copied into your notebook.
Graph of the function and its derivative
The blue curve is the function f(x), the orange one is the derivative f′(x). Green background: f′(x) > 0 and the function is increasing. Red background: f′(x) < 0 and the function is decreasing. The points where the derivative equals zero are marked on the graph — these are the extrema (peaks and valleys).
What a derivative is — three meanings
Geometric meaning
The derivative f′(x₀) is the slope of the tangent line to the graph at the point x₀, that is, the tangent of its angle of inclination. A large derivative means a steep rise, a negative one means a descent, zero means the tangent is horizontal (a peak or a valley). That is exactly why the sign of the derivative tells you whether the function is increasing or decreasing.
Physical meaning
If a function describes the position of a body as a function of time s(t), then the derivative s′(t) is the instantaneous velocity, and the second derivative s″(t) is the acceleration. In general the derivative is the rate of change of a quantity: how fast y changes as x changes. This is the original meaning for which Newton introduced the derivative.
Algebraic meaning
Formally the derivative is the limit of the ratio of increments: f′(x) = lim (f(x+h) − f(x)) / h as h → 0. We take a tiny step h, see how much the function changed, and divide by the length of the step. As the step approaches zero, the ratio approaches the exact rate of change at the point.
The derivative and extrema
At maximum and minimum points the tangent is horizontal, so the derivative there equals zero. This is the main technique for finding largest and smallest values: set the derivative to zero and find the roots. If, when passing through such a point, the derivative changes sign from “+” to “−”, it is a maximum; from “−” to “+”, a minimum.
Table of derivatives
The core formulas — the same ones used in the breakdown above. C is a constant, a > 0.
| f(x) | f′(x) |
|---|---|
| C | 0 |
| xⁿ | n·xⁿ⁻¹ |
| 1/x | −1/x² |
| √x | 1/(2√x) |
| eˣ | eˣ |
| aˣ | aˣ·ln a |
| ln x | 1/x |
| loga x | 1/(x·ln a) |
| sin x | cos x |
| cos x | −sin x |
| tg x | 1/cos²x |
| ctg x | −1/sin²x |
| arcsin x | 1/√(1 − x²) |
| arccos x | −1/√(1 − x²) |
| arctg x | 1/(1 + x²) |
📚 Rules of differentiation
The derivative of a sum equals the sum of the derivatives: (u + v)′ = u′ + v′. A constant factor is taken outside: (C·u)′ = C·u′.
Product rule: (u·v)′ = u′·v + u·v′. Quotient rule: (u/v)′ = (u′·v − u·v′) / v².
The chain rule for a composite function: first differentiate the outer function, then multiply by the derivative of the inner one. For example, (sin 2x)′ = cos(2x)·2.