Derivative Calculator

What this derivative calculator does — and what it doesn't

Type any single-variable expression in x — polynomials, trig functions, logs, exponentials, roots, or a mix — and this tool differentiates it symbolically using the same rules you'd apply by hand: power rule, product rule, quotient rule and chain rule. It returns the 1st, 2nd and 3rd derivative, plus a full step-by-step trace showing which rule fired at every sub-expression.

Honest scope: this is a single-variable elementary-function differentiator. It does not compute integrals, limits, partial derivatives, or solve differential equations — those need a different kind of tool entirely. If you need f'(x) or f''(x) for a function built from +, −, *, /, ^, and standard functions (sin, cos, tan, sec, csc, cot, ln, log, exp, sqrt, asin, acos, atan), this covers it.

The four core differentiation rules, with a worked example each

Power rule — d/dx[xⁿ] = n·xⁿ⁻¹. For f(x) = x³: f'(x) = 3x^2.

Product rule — d/dx[u·v] = u'v + uv'. For f(x) = x²sin(x) (u = x², v = sin(x)): f'(x) = 2x * sin(x) + x^2 * cos(x).

Quotient rule — d/dx[u/v] = (u'v − uv')/v². For f(x) = (x+1)/(x−3): f'(x) = (x - 3 - (x + 1)) / (x - 3)^2.

Chain rule — d/dx[f(g(x))] = f'(g(x))·g'(x). For f(x) = sin(x²) (outer sin, inner x²): f'(x) = 2x * cos(x^2). The chain rule is also what handles compound cases like f(x) = e^(2x)·cos(x): f'(x) = 2e^(2x) * cos(x) - e^(2x) * sin(x) (product rule on the outside, chain rule inside e^(2x)).

Higher-order derivatives: what f''(x) and f'''(x) actually tell you

The second derivative f''(x) is the derivative of the derivative — it measures how the slope itself is changing, which is why it's the standard tool for concavity (f'' > 0 means the curve bends upward) and, in physics, for acceleration when f(x) is a position function. The third derivative f'''(x) measures how the concavity is changing (jerk, in a physical motion context).

Worked example: for f(x) = x⁴ − 3x², f''(x) = 12x^2 - 6, so f''(2) = 42 — a positive value, meaning the graph is concave up at x = 2.

Common mistakes this tool helps you catch

The most common error is applying the power rule where the chain rule is needed — e.g. treating d/dx[sin(x²)] as cos(x²) instead of 2x·cos(x²) (forgetting to multiply by the derivative of the inner function x²). Another frequent mix-up is the product rule vs. the chain rule: x²sin(x) is a product of two functions of x (needs the product rule), while sin(x²) is one function nested inside another (needs the chain rule) — they look superficially similar but require different rules entirely.

A third common mistake: forgetting that the quotient rule's numerator is not symmetric — it's u'v − uv', not u'v + uv' and not uv' − u'v. Getting the order or sign wrong flips the result. Type your function in and compare the step-by-step trace against your own working to catch exactly where a rule was mis-applied.