Derivative Rules

From the Limit Definition to Practical Rules

The limit definition explains what a derivative means. Repeating the full limit calculation for several functions produces the same algebraic steps. Derivative rules record the results of those repeated calculations without changing the underlying idea of instantaneous rate of change.

For each function, first identify how it is built. Then choose the matching derivative rule and write each algebraic step clearly.

Constant Power and Linear Combination Rules

The definition of the derivative leads to a small set of rules that are faster to apply. A constant function, a power, and a sum of terms each get their own rule, and every rule reproduces the result the limit would have produced.

Constant and Power Functions

1. Constant rule: A constant function has a horizontal graph, so its slope is zero everywhere.

   $$f(x)=k \implies f'(x)=0$$

2. Power rule: For a constant $a$, multiply by the exponent, then reduce the exponent by one.

   $$f(x)=ax^n \implies f'(x)=anx^{n-1}$$

   This formula holds for every integer $n$ wherever $x^n$ is defined. For a general real exponent, it holds on $x>0$ and on any larger real domain where the power function is defined and differentiable.

Linear Combinations

Let $u(x)$ and $v(x)$ be differentiable functions.

1. Constant multiple: A constant factor stays in front of the derivative.

   $$f(x)=k\,u(x) \implies f'(x)=k\,u'(x)$$

2. Sum and difference: Differentiate each term separately and keep the operation between them.

   $$\begin{aligned} f(x) &=u(x)\pm v(x) \\ &\implies f'(x) \\ &=u'(x)\pm v'(x) \end{aligned}$$

Product Quotient and Chain Rules

Some expressions cannot be differentiated term by term. Each rule exists because the derivative of a combination is not the combination of the derivatives. In a product you differentiate one factor while the other stays fixed and add the two results, a quotient introduces a denominator, and a composition multiplies the outer derivative by the inner one.

Product Rule

For a product, differentiating both factors and multiplying the results is not enough. Differentiate one factor at a time while leaving the other unchanged, then add the two terms.

Quotient Rule

For a quotient, the order in the numerator matters. The denominator $v(x)$ must also be nonzero wherever the original function is defined.

Chain Rule

A composite function applies one function inside another. Differentiate the outer function at the unchanged inner input, then multiply by the derivative of that inner function.

Applying the Chain Rule to a Composite Function

Differentiate $f(x)=(x^2+1)^3$.

The outer function raises its input to the third power. The inner function is $u(x)=x^2+1$. Apply the power rule to the outer function, then multiply by the derivative of the inner function.

The factor $2x$ is essential. Leaving it out would mean differentiating only the outer function and stopping before the chain rule is complete.

Practice

Each problem needs one or more derivative rules combined. Decide on the rule before writing the derivative. Write the rule you chose next to the first line, so a wrong rule shows up before the algebra grows long.

1. Differentiate $h(x)=(2x-1)^4$.

Worked Solution

1. The outer function is a fourth power, while the inner function is $u(x)=2x-1$ with derivative $u'(x)=2$.

   $$h'(x)=4[u(x)]^3u'(x)$$

   $$h'(x)=8(2x-1)^3$$

   The result includes both parts of the composition: the derivative of the fourth power and the derivative of its inner linear function.