CALTRX

Derivative Calculator

Find derivatives of polynomial, trig, exponential, and log functions. See the complete solution with step-by-step working and formula explanations.

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Tips & Notes

  • ✓Power rule: bring exponent down, reduce by 1. d/dx[x⁷]=7x⁶. Works for fractions too: d/dx[x^(1/2)]=x^(−1/2)/2.
  • ✓Chain rule: derivative of outer function × derivative of inner. d/dx[e^(3x)]=e^(3x)×3=3e^(3x).
  • ✓Product rule memory: first × derivative of second + second × derivative of first.
  • ✓d/dx[constant]=0. d/dx[5x³]=15x². Constants disappear when differentiated.
  • ✓Check with units: if f(x) is position (m), f(x) is velocity (m/s) — dividing by time unit.

Common Mistakes

  • ✗Forgetting to apply the chain rule for composite functions. d/dx[sin(3x)] = cos(3x) × 3 = 3cos(3x), not just cos(3x). The inner function 3x has derivative 3, which must multiply the outer derivative.
  • ✗Misapplying the product rule by multiplying derivatives instead. d/dx[x²·sin(x)] = x²·cos(x) + sin(x)·2x — not x²·cos(x) alone or 2x·cos(x). Both terms of the product rule must be computed and added.
  • ✗Differentiating the denominator incorrectly with the quotient rule. d/dx[f/g] = (g·f′ − f·g′) / g². The numerator subtracts, not adds: (g×f′) minus (f×g′). Reversing the order of subtraction flips the sign of the result.
  • ✗Treating a constant multiplied by a function as a product requiring the product rule. d/dx[5sin(x)] = 5·cos(x) — the constant multiplier stays. The product rule only applies when both factors contain the variable x.
  • ✗Applying the power rule to exponential functions. d/dx[x³] = 3x² (power rule, exponent is a constant). But d/dx[3ˣ] = 3ˣ·ln(3) (exponential rule, base is a constant). Confusing these gives completely wrong results.

Derivative Calculator Overview

The derivative measures the instantaneous rate of change of a function — how fast a quantity is changing at a specific moment rather than on average across an interval. It is the central operation of differential calculus and appears in every field involving optimization, motion, growth, or change: physics, economics, engineering, biology, and machine learning all depend on it.

The formal definition expresses the derivative as a limit — the slope of the tangent line found by letting two points on a secant line merge into one:

f'(x) = lim(h→0) [f(x + h) − f(x)] / h
In practice, derivatives are computed using algebraic rules rather than evaluating this limit directly. The power rule handles the most common case:
d/dx [xⁿ] = n × xⁿ⁻¹
EX: d/dx [x⁵] = 5x⁴ | d/dx [x²] = 2x | d/dx [x] = 1 | d/dx [7] = 0
The product rule is needed whenever two functions are multiplied together — it cannot be replaced by differentiating each factor separately:
d/dx [f × g] = f' × g + f × g'
EX: d/dx [x² × sin(x)] = 2x·sin(x) + x²·cos(x)
The chain rule handles composite functions — any situation where one function is nested inside another:
d/dx [f(g(x))] = f'(g(x)) × g'(x)
EX: d/dx [sin(x²)] → outer function is sin, inner is x² → cos(x²) × 2x
Derivatives translate the abstract question "how fast is this changing?" into a concrete calculation applicable to any differentiable function. The chain rule, product rule, and quotient rule together handle virtually every function you will encounter in applied mathematics — they are the grammar of differential calculus, not special cases. Critical points where f′(x) = 0 or f′(x) is undefined are candidates for local maxima, minima, and inflection points. The second derivative test distinguishes them: f″ > 0 at a critical point means a local minimum; f″ < 0 means a local maximum; f″ = 0 is inconclusive. This combination — find critical points with f′, classify them with f″ — solves optimization problems across physics, engineering, economics, and every field where something is being maximized or minimized.

Frequently Asked Questions

The derivative f'(x) measures the instantaneous rate of change of f at point x — the slope of the tangent line at that exact location. If f(x) is position, f'(x) is velocity. If f(x) is revenue, f'(x) is marginal revenue: how much revenue changes per one additional unit sold. For f(x) = x squared, f'(x) = 2x, so at x = 3 the slope is 6 and at x = -1 the slope is -2.

The chain rule handles composite functions where one function is nested inside another: d/dx[f(g(x))] = f'(g(x)) times g'(x). Identify the outer function and the inner function, differentiate the outer while leaving the inner unchanged, then multiply by the derivative of the inner. Example: d/dx[sin(x squared)] = cos(x squared) times 2x = 2x cos(x squared). Every time you see an expression raised to a power, inside a trig function, or inside an exponential, the chain rule applies.

When two functions are multiplied together, differentiate using: d/dx[u times v] = u' times v + u times v'. Apply this whenever two distinct functions are multiplied, without trying to simplify first. Example: d/dx[x squared times sin(x)] = 2x times sin(x) + x squared times cos(x). A common error is to differentiate each factor separately and multiply the derivatives — that is wrong. The product rule adds two terms, not multiplies them.

Set f'(x) = 0 and solve for x to find critical point candidates. Then classify using the second derivative: if f''(x) > 0 the point is a local minimum; if f''(x) < 0 it is a local maximum; if f''(x) = 0 the test is inconclusive. Example: f(x) = x cubed minus 3x. f'(x) = 3x squared minus 3 = 0 gives x = 1 and x = -1. f''(x) = 6x. f''(1) = 6 > 0: local minimum at x = 1. f''(-1) = -6 < 0: local maximum at x = -1.

Implicit differentiation finds dy/dx when y is not isolated on one side of the equation. Differentiate both sides with respect to x, applying the chain rule to every term containing y by multiplying by dy/dx, then solve for dy/dx. Example: for the circle x squared + y squared = 25, differentiate to get 2x + 2y(dy/dx) = 0, giving dy/dx = -x/y. At point (3, 4) the slope is -3/4. Use this for any curve defined by an equation where separating y is difficult or impossible.

The second derivative f''(x) is the derivative of f'(x) — the rate of change of the rate of change. For position functions it gives acceleration. In curve analysis, f'' > 0 means the curve is concave up (like a cup) and f'' < 0 means concave down (like a cap). Inflection points occur where f'' changes sign — the curve switches from concave up to concave down or vice versa. Third and higher derivatives appear in Taylor series approximations and in physics: the third derivative of position with respect to time is called jerk.