Calculus

Furthermore

Official Guidance, clarification and syllabus links:

Examples: \(f(x)=e^{(x^2+2)}, \quad f(x)=\sin(3x-1) \)

Link to: composite functions (SL2.5).

Formula Booklet:

Derivative of \(\sin{x}\)	\( f(x)= \sin x \quad \Rightarrow \quad f'(x) = \cos x \)
Derivative of \(\cos{x} \)	\( f(x)= \cos x \quad \Rightarrow \quad f'(x) = i\sin x \)
Derivative of \(e^x\)	\( f(x)= e^x \quad \Rightarrow \quad f'(x) = e^x \)
Derivative of \(\ln x\)	\( f(x)= \ln x \quad \Rightarrow \quad f'(x) = \dfrac{1}{x} \)
Chain rule	\( y=g(u), \text{ where } u = f(x) \Rightarrow \quad \dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx} \)
Product rule	\( y=uv \quad \Rightarrow \quad \dfrac{dy}{dx} = u\dfrac{dv}{dx} + v\dfrac{du}{dx} \)
Quotient rule	\( y=\dfrac{u}{v} \quad \Rightarrow \quad \dfrac{dy}{dx} = \dfrac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \)

Examples

1. Chain Rule:
The chain rule is used when differentiating composite functions. If we have a function \( y = f(u) \) and \( u = g(x) \), then the derivative of \( y \) with respect to \( x \) is given by:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

Example: Differentiate \( y = \sin(3x^2) \) with respect to \( x \).

Let \( u = 3x^2 \). Then, \( \frac{du}{dx} = 6x \) and \( \frac{dy}{du} = \cos(u) \). Using the chain rule:

$$ \frac{dy}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2) $$

2. Product Rule:
The product rule is used when differentiating the product of two functions. If \( y = u \cdot v \) where both \( u \) and \( v \) are functions of \( x \), then the derivative of \( y \) with respect to \( x \) is:

$$ \frac{dy}{dx} = u' \cdot v + u \cdot v' $$

Example: Differentiate \( y = x^2 \cdot \ln(x) \) with respect to \( x \).

Using the product rule:

$$ \frac{dy}{dx} = 2x \cdot \ln(x) + x^2 \cdot \frac{1}{x} = 2x \ln(x) + x $$

3. Quotient Rule:
The quotient rule is used when differentiating the quotient of two functions. If \( y = \frac{u}{v} \) where both \( u \) and \( v \) are functions of \( x \) and \( v \neq 0 \), then the derivative of \( y \) with respect to \( x \) is:

$$ \frac{dy}{dx} = \frac{u' \cdot v - u \cdot v'}{v^2} $$

Example: Differentiate \( y = \frac{x^2}{\sin(x)} \) with respect to \( x \).

Using the quotient rule:

$$ \frac{dy}{dx} = \frac{2x \cdot \sin(x) - x^2 \cdot \cos(x)}{\sin^2(x)} $$

This video on the Basics of Differentiation is from Revision Village and is aimed at students taking the IB Maths Standard level course

Here's a 'deep fake' video featuring the images of Arnold Schwarzenegger, Ice Spice and Mr Beast explaining the chain rule for differentiation. While the humans might be fake the maths is real and correct.

If you use a TI-Nspire GDC there are instructions useful for this topic.