Basic Differentiation Rules

Differentiation rules are compressed first-principles arguments that let us focus on structure and interpretation.

Derivative notation

Equivalent notations include

Each represents the derivative function: the instantaneous rate of change of the output with respect to the input.

Constant rule

For any constant $c$,

A constant function is a horizontal line, so its slope is zero everywhere.

Power rule

For real exponents where the function and derivative are defined,

Multiply by the exponent, then reduce the exponent by one. For example,

Constant multiple rule

Constants pass through differentiation:

Thus

The multiplier changes vertical scale and therefore scales every tangent slope by the same amount.

Sum and difference rules

Differentiate term by term:

This makes polynomial differentiation straightforward once like terms and powers are clear.

Worked example: differentiate a polynomial

Negative and fractional exponents

Rewrite radicals and reciprocals as powers when helpful:

Then

for $x>0$. The derivative's domain may be narrower than the original function's domain.

Exponential derivatives

For the natural exponential,

More generally,

When the exponent is more than $x$, the chain rule is also required.

Tangent and normal lines

At $x=a$, tangent slope is $m_t=f'(a)$. Use point $(a,f(a))$:

When $f'(a)\ne0$, a normal line perpendicular to the tangent has slope $m_n=-1/f'(a)$.

Motion interpretation

If position is $s(t)$, velocity is $v(t)=s'(t)$ and acceleration is $a(t)=v'(t)=s''(t)$. Positive velocity means position increases; negative velocity means it decreases. Speed is $|v(t)|$.

Units change with each derivative: metres, metres per second, then metres per second squared.

Choose rules by structure

A sum is differentiated term by term. A product of nonconstant functions requires the product rule; a quotient requires the quotient rule; a function inside another requires the chain rule.

Simplifying first may reduce work, but preserve the original domain.

Common mistakes

Reducing the exponent without multiplying by it. The power rule has both actions.

Differentiating a constant as itself. Its derivative is zero.

Using the power rule on a product without the product rule. Identify structure first.

Forgetting domain changes with negative or fractional powers. State valid inputs.

Using $f'(a)$ as the tangent point's height. The point uses $f(a)$; the slope uses $f'(a)$.

Quick self-check

• Has the expression been simplified into clear terms and factors?
• Which differentiation rule matches its outer structure?
• Did constants and powers receive correct derivatives?
• Is the derivative domain stated when necessary?
• Does a tangent slope agree with the graph's local direction?
• Are contextual units correct?

Related topics

• Derivative from First Principles
• Product Rule
• Quotient Rule
• Chain Rule