A derivative measures how a function changes as its input changes — it's the instantaneous rate of change or the slope of the function at any given point. Derivatives are the foundation of calculus and essential for optimization, physics, and engineering.
Basic Differentiation Rules
Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹
For any real number n. Example: d/dx(x³) = 3x²
Constant Rule: d/dx(c) = 0
The derivative of any constant is zero.
Sum/Difference Rule: d/dx(f ± g) = f' ± g'
Differentiate each term separately, then combine.
Constant Multiple Rule: d/dx(c·f) = c·f'
Constants factor out of the derivative.
Trigonometric Derivatives
d/dx(sin x) = cos x
Derivative of sine is cosine.
d/dx(cos x) = −sin x
Derivative of cosine is negative sine.
d/dx(tan x) = sec²(x)
Derivative of tangent is secant squared.
Exponential & Logarithmic Derivatives
d/dx(eˣ) = eˣ
The exponential function is its own derivative.
d/dx(ln x) = 1/x
Derivative of the natural logarithm.
How to Differentiate Step by Step
1
Identify the function type: Is it a polynomial, trigonometric, exponential, or a combination?
2
Break into terms: Separate the function into individual terms using the sum/difference rule.
3
Apply the appropriate rule: Use power rule for xⁿ, trig rules for sin/cos/tan, or exponential rules for eˣ/ln.
4
Simplify: Combine like terms and simplify the final expression.
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Power Rule
Differentiate polynomial functions using the power rule: d/dx(xⁿ) = n·xⁿ⁻¹. Handles positive, negative, and fractional exponents.
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Trigonometric Derivatives
Compute derivatives of sin(x), cos(x), and tan(x) instantly. Knows all the standard trig differentiation rules.
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Exponential & Log Rules
Differentiate eˣ, ln(x), and sqrt(x). Supports exponential functions and natural logarithms with ease.
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Step-by-Step Solutions
See every differentiation step explained clearly — from identifying terms to applying rules and simplifying the final result.
A derivative measures the rate at which a function changes at any given point. In geometric terms, the derivative of a function at a point equals the slope of the tangent line to the function's graph at that point. If you think of a function as a curve, the derivative tells you how steep the curve is at each point along its path.
Derivatives are fundamental to calculus and have countless applications across mathematics, physics, engineering, economics, and data science. They are used to find maximum and minimum values (optimization), model rates of change (velocity, acceleration), approximate functions (linearization), and solve differential equations that describe natural phenomena.
The formal definition of a derivative is the limit of the difference quotient:
f'(x) = lim[h→0] (f(x+h) − f(x)) / h
This limit gives the instantaneous rate of change of f at a point x.
In practice, instead of evaluating this limit every time, we use differentiation rules — like the power rule, product rule, chain rule, and trigonometric rules — to compute derivatives quickly and systematically.
Differentiation Rules Explained
The Power Rule
The power rule is the most commonly used differentiation rule. It states that for any function of the form xⁿ, the derivative is n·xⁿ⁻¹. For example, the derivative of x⁴ is 4x³, and the derivative of √x (which is x^½) is ½·x^−½ = 1/(2√x).
The Sum and Constant Multiple Rules
The sum rule tells us that the derivative of a sum of functions is the sum of their individual derivatives: d/dx(f(x) + g(x)) = f'(x) + g'(x). The constant multiple rule says that constants factor out: d/dx(c·f(x)) = c·f'(x). Together, these rules allow us to differentiate polynomials term by term.
Trigonometric Derivatives
Each trigonometric function has a specific derivative that you can learn and apply. The derivatives of the three basic trig functions are: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, and d/dx(tan x) = sec²(x). These are derived from the limit definition and the geometric properties of the unit circle.
Exponential and Logarithmic Derivatives
The exponential function eˣ is unique in that its derivative is itself: d/dx(eˣ) = eˣ. The natural logarithm has the derivative d/dx(ln x) = 1/x. These rules are essential for modeling growth, decay, and many natural processes.
Applications of Derivatives
Derivatives appear in virtually every quantitative field. Here are some of the most important applications:
Physics: Velocity is the derivative of position with respect to time. Acceleration is the derivative of velocity. Derivatives describe motion, force, energy, and fluid dynamics.
Engineering: Derivatives are used in control systems, signal processing, structural analysis, and electrical circuit design to model how systems respond to changes.
Economics & Finance: Marginal cost and marginal revenue are derivatives of cost and revenue functions. Derivatives are also used in options pricing (the Black-Scholes model) and risk analysis.
Data Science & Machine Learning: Gradient descent — the core optimization algorithm in neural networks — uses derivatives (gradients) to minimize loss functions and improve model accuracy.
Biology & Medicine: Derivatives model population growth rates, drug concentration changes in the bloodstream, and the spread of diseases.
Frequently Asked Questions
What is a derivative in simple terms?
A derivative measures how a function changes as its input changes. Imagine you're driving — the derivative of your position with respect to time is your speed. It tells you how fast you're going at any instant. Mathematically, the derivative f'(x) gives the slope of the tangent line to the curve y = f(x) at point x.
How do I use the Power Rule for differentiation?
The power rule states: d/dx(xⁿ) = n·xⁿ⁻¹. To use it, bring the exponent down as a coefficient, then reduce the exponent by 1. For example, d/dx(x⁵) = 5x⁴, d/dx(x²) = 2x, and d/dx(x) = 1 (since x = x¹). This works for any real exponent, including fractions and negative numbers: d/dx(√x) = d/dx(x^½) = ½·x^−½ = 1/(2√x).
What is the derivative of sin(x) and cos(x)?
The derivative of sin(x) is cos(x), and the derivative of cos(x) is −sin(x). These are the two most fundamental trigonometric derivatives. From these, other trig derivatives can be derived: d/dx(tan x) = sec²(x), d/dx(cot x) = −csc²(x), d/dx(sec x) = sec(x)·tan(x), and d/dx(csc x) = −csc(x)·cot(x).
What is the difference between a derivative and an integral?
A derivative measures the rate of change of a function, while an integral measures the accumulation of a quantity. They are inverse operations — the Fundamental Theorem of Calculus states that differentiation and integration undo each other. If you integrate a derivative, you get back the original function (plus a constant). If you differentiate an integral, you get back the original function.
How do I differentiate eˣ and ln(x)?
The derivative of eˣ is simply eˣ — the exponential function is its own derivative, which is a unique and important property. The derivative of the natural logarithm ln(x) is 1/x. For a general exponential aˣ, the derivative is aˣ·ln(a). For a general logarithm logₐ(x), the derivative is 1/(x·ln(a)).
What is the derivative of a constant?
The derivative of any constant is zero. This makes intuitive sense because a constant doesn't change — its rate of change is always zero. For example, if f(x) = 5, then f'(x) = 0. Similarly, the derivative of 100, π, e, or any other constant value is always zero.
⚠️ Important Note: This Derivative Calculator is for educational and reference purposes. While the differentiation rules are mathematically verified, results should be double-checked for critical academic or professional work. Always consult your textbook or instructor for verification of calculus results in high-stakes contexts.