IB Mathematics Applications & Interpretation Topic AHL 5.9—differentiating Standard Functions and Derivative Rules Revision Notes (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.9—differentiating Standard Functions and Derivative Rules Revision Notes (HL) | RevisionDojo

Written by official IB examiners

Derivatives of Standard Functions

Trigonometric Functions

The derivatives of the basic trigonometric functions are fundamental in calculus:

$\frac{d}{dx}(\sin x) = \cos x$
$\frac{d}{dx}(\cos x) = -\sin x$
$\frac{d}{dx}(\tan x) = \sec^2 x$

Example

Let's differentiate $f(x) = \sin(2x)$:

Using the chain rule (which we'll explore later), we get: $f'(x) = \cos(2x) \cdot 2 = 2\cos(2x)$

Note

The derivatives of inverse trigonometric functions, while not explicitly mentioned in the syllabus, are often useful in more advanced problems:

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$

Exponential and Logarithmic Functions

The exponential and natural logarithm functions have particularly elegant derivatives:

$\frac{d}{dx}(e^x) = e^x$
$\frac{d}{dx}(\ln x) = \frac{1}{x}$

Example

Differentiate $g(x) = e^{3x}$:

Again, using the chain rule: $g'(x) = e^{3x} \cdot 3 = 3e^{3x}$

Power Function

For the general power function:

$\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a rational number

This rule is incredibly versatile, applying to integer, fractional, and negative exponents.

Example

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

$h'(x) = -\frac{1}{2}x^{-3/2}$

Common Mistake

Students often forget that this rule applies to negative and fractional exponents. Remember, $x^{-1} = \frac{1}{x}$ and $x^{1/2} = \sqrt{x}$.

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Derivatives of Standard Functions

Trigonometric Functions

The derivatives of the basic trigonometric functions are fundamental in calculus:

$\frac{d}{dx}(\sin x) = \cos x$
$\frac{d}{dx}(\cos x) = -\sin x$
$\frac{d}{dx}(\tan x) = \sec^2 x$

Example

Let's differentiate $f(x) = \sin(2x)$:

Using the chain rule (which we'll explore later), we get: $f'(x) = \cos(2x) \cdot 2 = 2\cos(2x)$

Note

The derivatives of inverse trigonometric functions, while not explicitly mentioned in the syllabus, are often useful in more advanced problems:

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$

Exponential and Logarithmic Functions

The exponential and natural logarithm functions have particularly elegant derivatives:

$\frac{d}{dx}(e^x) = e^x$
$\frac{d}{dx}(\ln x) = \frac{1}{x}$

Example

Differentiate $g(x) = e^{3x}$:

Again, using the chain rule: $g'(x) = e^{3x} \cdot 3 = 3e^{3x}$

Power Function

For the general power function:

$\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a rational number

This rule is incredibly versatile, applying to integer, fractional, and negative exponents.

Example

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

$h'(x) = -\frac{1}{2}x^{-3/2}$

Common Mistake

Students often forget that this rule applies to negative and fractional exponents. Remember, $x^{-1} = \frac{1}{x}$ and $x^{1/2} = \sqrt{x}$.

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

IB Mathematics Applications & Interpretation AHL 5.9—differentiating Standard Functions and Derivative Rules Notes

Derivatives of Standard Functions

Trigonometric Functions

Exponential and Logarithmic Functions

Power Function

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Derivatives of Standard Functions

Trigonometric Functions

Exponential and Logarithmic Functions

Power Function

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics