Free lesson · Differential calculus

Core derivative rules: constants, powers and sums

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Start with the idea

A derivative is a local slope.
Start with one constant, one power and one sum before combining functions.

Symbols, units & horizon

x: dimensionless input in this example
f(x): dimensionless output
f′(x): output change per input change locally
d/dx: differentiate with respect to x
n: positive integer power
a: constant multiplier
g: another differentiable function
h: nonzero input increment before the limit

When and why to use this

Use these rules to differentiate a simple cost curve or loss function before tackling product, quotient and chain rules.

Constant rule: $\frac{d}{d x} 7 = 0$ . The output never changes.
Identity rule: $\frac{d}{d x} x = 1$ . Output and input change equally.
Power rule: $\frac{d}{d x} x^{n} = n x^{n - 1}$ for positive integer n.
Constant multiple: $(a f)^{'} = a f^{'}$ when a is fixed.
Sum rule: $(f + g)^{'} = f^{'} + g^{'}$ .
Apply each rule to one term. Leave products and nested functions for the next lesson.

f (x) = 3 x^{2} + 2 x + 7, f^{'} (x) = 6 x + 2

Core rule · definition and worked arithmetic

Core derivative rules: constants, powers and sums

Constant: the change quotient is $\frac{(7 - 7)}{h} = 0$ . Identity: $\frac{((x + h) - x)}{h} = 1$ .
Square: expand $\frac{((x + h)^{2} - x^{2})}{h} = 2 x + h$ ; its limit is $2 x$ .
For a positive integer power, the binomial expansion starts $(x + h)^{n} = x^{n} + n x^{n - 1} h + \dots$ . Higher terms contain at least h²; after dividing by h they tend to zero.
Split the change quotient of a sum into separate quotients; fixed multipliers can be taken outside the limit.
Apply these rules: $3 (2 x) + 2 (1) + 0 = 6 x + 2$ .

Work it by hand

At x=2, the derivative is 6×2+2=14. The function value is 3×4+4+7=23; value and slope are different quantities.

Use the rule

Name the inputs and units.
Work the small example by hand.
Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Further reading: OpenStax · Differentiation rules · textbook checked 12 September 2026 ↗

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def polynomial(x): return 3*x*x+2*x+7

def polynomial_derivative(x): return 6*x+2

def positive_integer_power_derivative(x, n):
    if not isinstance(n,int) or n < 1: raise ValueError("Positive integer power required")
    return n*x**(n-1)

print(polynomial(2), polynomial_derivative(2))

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