AkCalculators

Derivative —

Differentiation is one of the central operations in calculus — it measures how a quantity changes with respect to another. If a function f(x) describes a position, its derivative f'(x) describes velocity; if f(x) is cost, f'(x) captures marginal cost. The derivative gives both a local linear approximation to a function and a measure of instantaneous change. This guide covers the foundational rules, intuition, worked examples, and practical applications you can use to become comfortable with derivatives.

What is the derivative?

Informally, the derivative of f at a point x measures the slope of the tangent line to the curve y = f(x) at that point. Formally, the derivative is defined as a limit (when it exists):

Basic rules you must know

A handful of rules are used repeatedly:

• Constant rule: d/dx[c] = 0.
• Power rule: d/dx[x^n] = n·x^{n-1}. This is especially useful for polynomials and rational powers when n is an integer.
• Constant multiple rule: d/dx[a·f(x)] = a·f'(x).
• Sum rule: d/dx[f(x)+g(x)] = f'(x) + g'(x).
• Product rule: d/dx[f·g] = f'·g + f·g'.
• Quotient rule: d/dx[f/g] = (f'·g - f·g') / g^2.
• Chain rule: for composition f(g(x)), d/dx[f(g(x))] = f'(g(x))·g'(x). This rule is essential whenever one function appears inside another — for example, sin(3x) or e^{2x}.\li>

Power rule and polynomials

Polynomials are sums of power terms like a_n x^n + ... + a_1 x + a_0. Because differentiation distributes across sums and constants, we can differentiate polynomials easily: apply the power rule to each term and add results. For example, if f(x) = 4x^3 - 2x + 7, then f'(x) = 12x^2 - 2. The constant 7 disappears and each power drops by one while multiplying by the original exponent.

Trigonometric, exponential and logarithmic derivatives

Several elementary functions have standard derivatives:

• d/dx[sin x] = cos x
• d/dx[cos x] = -sin x
• d/dx[tan x] = sec^2 x
• d/dx[e^x] = e^x
• d/dx[ln x] = 1/x (x>0)

When these functions appear with linear inner terms (for example sin(kx) or e^{kx}) the chain rule tells us to multiply by the derivative of the inner linear function (k). For sin(3x), derivative is 3·cos(3x).

Chain, product and quotient rules — a few examples

The chain rule is used when functions are nested. Example: differentiate h(x) = (3x^2 + 1)^5. First think of the outer function as u^5 with u = 3x^2 + 1. Then h'(x) = 5·u^4 · u' = 5(3x^2 + 1)^4 · 6x = 30x(3x^2 + 1)^4. The product and quotient rules handle products or ratios directly; they are particularly important when functions multiply or divide each other.

Interpreting derivatives

The derivative value at a point tells whether the function is increasing (positive derivative) or decreasing (negative derivative) nearby. Zeros of the derivative (critical points) often indicate maxima, minima, or inflection points, though second-derivative tests or further analysis are usually required to classify them.

Practical applications

Derivatives appear everywhere: physics (velocity, acceleration), engineering (stress-strain relationships), economics (marginal cost and revenue), biology (rates of reaction), and machine learning (gradients). Learning to compute derivatives quickly and understand their meaning is essential for modeling and optimizing real-world systems.

How to use this calculator

Use Polynomial Mode for straightforward polynomials — the tool applies the power rule term-by-term and shows each intermediate step. Switch to Function Mode for sin, cos, tan, e^x, ln and for simple chain-rule patterns like sin(3x) or e^(2x). For products, quotients, or very complex symbolic expressions, this tool gives guidance and best-effort steps; for full symbolic manipulation consider a CAS (computer algebra system).

Next steps

After mastering single-variable derivatives, next learn higher derivatives, implicit differentiation, and techniques for optimization with constraints. Multivariable calculus introduces partial derivatives and gradients — a fundamental toolset for modern data science and engineering.

Differentiation rewards pattern recognition and practice. Use this calculator to check work, study the steps, and then try to reproduce solutions without help. With time you will build fluency and the rules will become intuitive tools rather than memorized formulas.