Derivatives of Common Functions

Using the limit definition and the angle-addition identity sin(x+h) = sin x cos h + cos x sin h, we get: d/dx[sin x] = lim(h→0) [sin(x+h) − sin x] / h = lim(h→0) [sin x (cos h − 1)/h + cos x (sin h)/h]. Two fundamental trigonometric limits tell us lim(h→0)(sin h)/h = 1 and lim(h→0)(cos h − 1)/h = 0. Substituting gives d/dx[sin x] = cos x.

A similar derivation uses cos(x+h) = cos x cos h − sin x sin h. Applying the same two limits: d/dx[cos x] = lim(h→0) [cos x (cos h − 1)/h − sin x (sin h)/h] = cos x · 0 − sin x · 1 = −sin x. The negative sign is crucial: the cosine function decreases where the sine function is increasing. Notice also that differentiating sin x twice returns −sin x, and four times returns sin x — a cyclic pattern of period 4.

The natural exponential function eˣ has the remarkable property of being its own derivative: d/dx[eˣ] = eˣ. This can be shown via the limit definition using the fact that lim(h→0)(eʰ − 1)/h = 1, which is equivalent to the definition of e itself. The number e ≈ 2.71828 is defined precisely so that this derivative holds. For a general base: d/dx[aˣ] = aˣ ln a, where the extra factor ln a equals 1 only when a = e, confirming e is the 'natural' base.

Since ln x is the inverse of eˣ, we can derive its derivative using implicit differentiation. Let y = ln x, so eʸ = x. Differentiating both sides with respect to x: eʸ · (dy/dx) = 1, giving dy/dx = 1/eʸ = 1/x. Therefore d/dx[ln x] = 1/x, valid for x > 0. This result is especially important in integration: ∫(1/x)dx = ln|x| + C, the only power-rule exception where the integral of xⁿ would otherwise break down at n = −1.

The four rules together are: d/dx[sin x] = cos x; d/dx[cos x] = −sin x; d/dx[eˣ] = eˣ; d/dx[ln x] = 1/x. These combine with the chain rule for more complex functions. For example: d/dx[sin(3x)] = 3cos(3x); d/dx[e^(x²)] = 2x·e^(x²); d/dx[ln(5x)] = 1/x. Notice that for any constant c, d/dx[ln(cx)] = 1/x since d/dx[ln c + ln x] = 0 + 1/x = 1/x — the constant inside the logarithm vanishes upon differentiation.

• \sin x — Sine function (dimensionless): Fundamental trigonometric function; ratio of the opposite side to the hypotenuse in a right triangle, extended to all real numbers via the unit circle.
• e^x — Natural exponential function (dimensionless): Exponential function with base e ≈ 2.71828; the unique function (up to scaling) that equals its own derivative.
• \ln x — Natural logarithm (dimensionless): The inverse function of eˣ; defined for x > 0. Its derivative 1/x is the key to integrating the reciprocal function.
• \frac{d}{dx} — Differentiation operator (per unit of x): The operator that takes a function and returns its derivative — the instantaneous rate of change with respect to x.

Problem. Find the derivative of f(x) = 3sin x − 2eˣ + 5ln x and evaluate f ′(1).

Answer. f ′(x) = 3cos x − 2eˣ + 5/x; f ′(1) ≈ 1.184

• The Limit Definition of the Derivative — All four derivative formulas on this page are rigorously proven using the limit definition lim(h→0)[f(x+h)−f(x)]/h. Understanding this foundation clarifies why the formulas work.
• The Chain Rule — The chain rule extends these four basic formulas to composite functions such as sin(x²), e^(3x), and ln(x²+1), making it the most important rule in differential calculus.
• Product and Quotient Rules — When two common functions are multiplied or divided — for example x·sin x or eˣ/ln x — the product and quotient rules are needed alongside the basic derivative formulas.
• Integration as Antidifferentiation — Each derivative formula here reverses into an integration formula: ∫cos x dx = sin x + C, ∫eˣ dx = eˣ + C, ∫(1/x)dx = ln|x| + C. The two topics are inseparable.
• Exponential Growth and Decay — The fact that d/dx[eˣ] = eˣ directly explains why exponential functions model growth and decay: the rate of change at any moment is proportional to the current value.