Basic Differentiation Rules

Core differentiation rules turn constants, powers, sums, and scalar multiples into instantaneous-rate formulas efficiently.

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Identify structure → choose rule → verify. A formula or convergence question is inspected for structure and conditions, routed to an appropriate calculus rule or test, and checked against units, endpoints, differentiation, or the stated hypotheses.

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

Derivative notation

Equivalent notations include

f'(x),\qquad y',\qquad \frac{dy}{dx},\qquad \frac{d}{dx}[f(x)].

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

Constant rule

For any constant $c$,

\frac{d}{dx}[c]=0.

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,

\frac{d}{dx}[x^n]=nx^{n-1}.

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

\frac{d}{dx}[x^5]=5x^4,qquad \frac{d}{dx}[x^{-2}]=-2x^{-3}.

Constant multiple rule

Constants pass through differentiation:

\frac{d}{dx}[cf(x)]=cf'(x).

Thus

\frac{d}{dx}[7x^4]=28x^3.

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

Sum and difference rules

Differentiate term by term:

\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x).

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

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 $f'(a)$.