Standard Derivatives: An Overview

Differentiation is the process of finding the derivative of a function , the instantaneous rate of change. Before applying the powerful rules of calculus, you need to know the derivatives of the core standard functions by heart. These form the building blocks for every more complex differentiation problem.

The key standard results you must know are:

Power Functions:
$\frac{d}{d x} (x^{n}) = n x^{n - 1}, n \in Q$

Trigonometric Functions:
$\frac{d}{d x} (sin x) = cos x, \frac{d}{d x} (cos x) = - sin x, \frac{d}{d x} (tan x) = sec^{2} x$

Exponential and Logarithmic Functions:
$\frac{d}{d x} (e^{x}) = e^{x}, \frac{d}{d x} (ln x) = \frac{1}{x}$

Note

All of these results appear in the IB Mathematics: AI HL formula booklet, but you should be so familiar with them that you rarely need to look them up. Speed in exams depends on internalising these.

Analogy

Think of these standard derivatives as vocabulary in a language. The rules (chain, product, quotient) are the grammar. You can't form sentences until you know your words.

The Power Rule

Power Rule: For any rational number $n$ , the derivative of $x^{n}$ is:
$\frac{d}{d x} (x^{n}) = n x^{n - 1}$

This rule is far more versatile than it first appears. It applies to:

Positive integer powers: $\frac{d}{d x} (x^{4}) = 4 x^{3}$
Negative powers: $\frac{d}{d x} (x^{- 2}) = - 2 x^{- 3}$
Fractional (rational) powers: $\frac{d}{d x} (x^{1/3}) = \frac{1}{3} x^{- 2/3}$

Warning

A very common mistake is forgetting to rewrite expressions before differentiating. You cannot directly apply the power rule to $\frac{1}{x ^{3}}$ or $4 x$ in their original forms , you must rewrite them as $x^{- 3}$ and $x^{1/4}$ first.

Example

Differentiate $h (x) = x^{- 1/2}$ :

Apply the power rule with $n = - \frac{1}{2}$ :
$h^{'} (x) = - \frac{1}{2} \cdot x^{- 1/2 - 1} = - \frac{1}{2} x^{- 3/2}$

This can also be written as $h^{'} (x) = - \frac{1}{2 x ^{3/2}}$ .

Differentiate $f (x) = \frac{3}{x ^{4}}$ :

Rewrite first: $f (x) = 3 x^{- 4}$

$f^{'} (x) = 3 \cdot (- 4) x^{- 5} = - 12 x^{- 5} = - \frac{12}{x ^{5}}$

Free preview

10 more sections in this topic

Standard Derivatives: An Overview

The key standard results you must know are:

Power Functions:
$\frac{d}{d x} (x^{n}) = n x^{n - 1}, n \in Q$

Trigonometric Functions:
$\frac{d}{d x} (sin x) = cos x, \frac{d}{d x} (cos x) = - sin x, \frac{d}{d x} (tan x) = sec^{2} x$

Exponential and Logarithmic Functions:
$\frac{d}{d x} (e^{x}) = e^{x}, \frac{d}{d x} (ln x) = \frac{1}{x}$

Note

All of these results appear in the IB Mathematics: AI HL formula booklet, but you should be so familiar with them that you rarely need to look them up. Speed in exams depends on internalising these.

Analogy

Think of these standard derivatives as vocabulary in a language. The rules (chain, product, quotient) are the grammar. You can't form sentences until you know your words.

The Power Rule

Power Rule: For any rational number $n$ , the derivative of $x^{n}$ is:
$\frac{d}{d x} (x^{n}) = n x^{n - 1}$

This rule is far more versatile than it first appears. It applies to:

Positive integer powers: $\frac{d}{d x} (x^{4}) = 4 x^{3}$
Negative powers: $\frac{d}{d x} (x^{- 2}) = - 2 x^{- 3}$
Fractional (rational) powers: $\frac{d}{d x} (x^{1/3}) = \frac{1}{3} x^{- 2/3}$

Warning

Example

Differentiate $h (x) = x^{- 1/2}$ :

Apply the power rule with $n = - \frac{1}{2}$ :
$h^{'} (x) = - \frac{1}{2} \cdot x^{- 1/2 - 1} = - \frac{1}{2} x^{- 3/2}$

This can also be written as $h^{'} (x) = - \frac{1}{2 x ^{3/2}}$ .

Differentiate $f (x) = \frac{3}{x ^{4}}$ :

Rewrite first: $f (x) = 3 x^{- 4}$

$f^{'} (x) = 3 \cdot (- 4) x^{- 5} = - 12 x^{- 5} = - \frac{12}{x ^{5}}$

Free preview

AHL 5.9—Differentiating standard functions and derivative rules

Standard Derivatives: An Overview

The Power Rule

10 more sections in this topic

AHL 5.9—Differentiating standard functions and derivative rules

Standard Derivatives: An Overview

The Power Rule

10 more sections in this topic