Derivative Calculator

How Derivatives Are Computed

A derivative measures the instantaneous rate of change of a function — the slope of its graph at any point. The formal definition is: f'(x) = lim(h→0) [f(x+h) − f(x)] / h. In practice, derivatives are computed using established rules rather than this limit every time. The Power Rule handles xⁿ (giving nxⁿ⁻¹). The Product Rule handles f·g (giving f'g+fg'). The Chain Rule handles compositions f(g(x)) (giving f'(g(x))·g'(x)). The Quotient Rule handles f/g (giving (f'g−fg')/g²).

Common Derivative Formulas

d/dx[xⁿ] = nxⁿ⁻¹ · d/dx[eˣ] = eˣ · d/dx[ln x] = 1/x · d/dx[sin x] = cos x · d/dx[cos x] = −sin x · d/dx[tan x] = sec²x · d/dx[arctan x] = 1/(1+x²) · d/dx[arcsin x] = 1/√(1−x²). These formulas combined with the four main rules allow differentiation of almost any expression.

Step-by-Step Derivative Examples

Example 1: f(x) = 3x⁴ − 2x² + 7x − 1 → Apply Power Rule to each term: f'(x) = 12x³ − 4x + 7.

Example 2: f(x) = x²·sin(x) → Product Rule: f'(x) = 2x·sin(x) + x²·cos(x).

Example 3: f(x) = sin(x²+1) → Chain Rule: f'(x) = cos(x²+1)·2x = 2x·cos(x²+1).

Example 4: f(x) = eˣ/(x+1) → Quotient Rule: f'(x) = [eˣ(x+1) − eˣ·1]/(x+1)² = eˣ·x/(x+1)².

When to Use Each Rule

The right approach depends on the structure of the function. Product Rule: two functions multiplied, neither a simple constant. Quotient Rule: a fraction where both numerator and denominator depend on x. Chain Rule: any composition — any function where the argument is not plain x. Power Rule: any term of the form xⁿ with constant n. Multiple rules often apply simultaneously: d/dx[sin(x²)·eˣ] requires both the Product Rule (sin(x²) times eˣ) and the Chain Rule (for sin(x²)).