Derivative Formula: Rules, List & Examples

The derivative formula, f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}f′(x)=limh→0hf(x+h)−f(x), measures a function's instantaneous rate of change at a single point, applied through standard rules — power, sum, product, quotient, and chain — together with the derivatives of trig, exponential, logarithmic, and inverse trig functions, with the chain rule and the product rule causing most student errors.

What the derivative formula is

The derivative formula measures how fast a function's output changes when its input changes by a tiny amount. It is the foundation of differential calculus. For a function f(x), the derivative is written as f′(x) or dfdx\dfrac{df}{dx}. The derivative is defined by the limit:

The Derivative Formula (First Principles):
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim\_{h \to 0} \dfrac{f(x+h) - f(x)}{h}

This is the formal definition. Most working calculus uses shortcut rules derived from it.

Variable Key

Symbol	Meaning
f(x)	The original function
f′(x)	The derivative of f(x) — its rate of change
dfdx	Leibniz notation for the same derivative
h	A tiny change in x, shrinking toward zero
lim⁡h→0	"As h approaches zero" — the limiting value

When To Use The Derivative Formula

Use a derivative whenever you need the rate of change at a single instant rather than over a stretch. Position changing into velocity. Velocity changing into acceleration. A drug concentration falling in the bloodstream. The slope of a curve at one specific point. Any time a problem asks "how fast," "how steep," "the rate at which," or "the marginal," a derivative is the tool.

The Full List Of Derivative Formulas

The list below is the working toolkit. Most calculus problems are solved by recognising which formula applies and substituting.

Basic derivative formulas

Function	Derivative
c	0
x	1
x^n	n⋅x^(n-1)
\sqrt{x}	1/(2√x)
1/x	-1/x^2

Trigonometric derivative formulas

Function	Derivative
sin⁡x	cos⁡x
cos⁡x	-sin⁡x
tan⁡x	sec²x
cot⁡x	-csc²x
sec⁡x	sec⁡x⋅tan⁡x
csc⁡x	-csc⁡x⋅cot⁡x

Exponential and logarithmic derivative formulas

Function	Derivative
e^x	e^x
a^x	a^x⋅ln⁡a
ln⁡x	1/x
logₐx	1/(x⋅ln⁡a)

Inverse trigonometric derivative formulas

Function	Derivative
sin⁡⁻¹x	1/√(1-x²)
cos⁡⁻¹x	-1/√(1-x²)
tan⁡⁻¹x	1/(1+x²)
cot⁡⁻¹x	-1/(1+x²)
sec⁡⁻¹x	1/
csc⁡⁻¹x	-1/

Differentiation Rules

Rules combine the basic formulas above when functions are added, multiplied, divided, or nested.

1. Constant Multiple Rule.
ddx[c⋅f(x)] = c⋅f′(x)

2. Sum and Difference Rule.
ddx[f(x)±g(x)] = f′(x)±g′(x)

3. Product Rule.
ddx[f(x)⋅g(x)] = f′(x)⋅g(x)+f(x)⋅g′(x)

4. Quotient Rule.
ddx[f(x)/g(x)] = (f′(x)⋅g(x)−f(x)⋅g′(x))/[g(x)]²

5. Chain Rule.
ddx[f(g(x)] = f′(g(x))⋅g′(x)

Intuitive Derivation - Where The Formula Comes From

The slope of the line connecting two points on a curve approaches the instantaneous rate of change, which is the derivative.

Worked Examples of Derivative Formula

Example 1: Derivative of f(x)=x⁴

Apply the power rule:
f′(x)=4⋅x³

Final answer: f′(x)=4x³.

Example 2: Derivative of f(x)=3x⁵−2x³+7x−9

Apply the sum/difference rule term by term.

Final answer: f′(x)=15x⁴−6x²+7.

Example 3: Derivative of f(x)=x²⋅sin⁡x

The correct tool is the product rule:
f′(x)=(2x)(sin⁡x)+(x²)(cos⁡x)

Final answer: f′(x)=2xsin⁡x+x²cos⁡x.

Example 4: Derivative of f(x)=sin⁡(x²)

The correct application is using the chain rule: f′(x)=2xcos⁡(x²)

Final answer: f′(x)=2xcos⁡(x²).

Common Mistakes to Avoid

Forgetting the chain rule on composite functions.
Misapplying the product rule.
Dropping the negative sign on ddx[cos⁡x].
Power rule confusion when the exponent is the variable.