IB Mathematics AI AHL The derivatives of sin x MAI Study Notes - New Syllabus
IB Mathematics AI AHL The derivatives of sin x MAI Study Notes
LEARNING OBJECTIVE
- The derivatives of standard functions.
Key Concepts:
- Common Derivatives
- Derivative Rules
- Related Rates of Change
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COMMON DERIVATIVES
Common Derivatives
Below are the standard derivatives of common functions:
| Function | Derivative |
|---|---|
| \( f(x) = x^n \) | \( f'(x) = nx^{n-1} \), where \( n \in \mathbb{Q} \) |
| \( f(x) = \sin x \) | \( f'(x) = \cos x \) |
| \( f(x) = \cos x \) | \( f'(x) = -\sin x \) |
| \( f(x) = \tan x \) | \( f'(x) = \sec^2 x \) (where \( x \ne \frac{\pi}{2} + n\pi \)) |
| \( f(x) = e^x \) | \( f'(x) = e^x \) |
| \( f(x) = \ln x \) | \( f'(x) = \frac{1}{x} \), \( x > 0 \) |
Example Differentiate each of the following:
▶️Answer/Explanation
|
DERIVATIVE RULES
Chain Rule
The chain rule is used to differentiate a function that is composed of two or more functions. If one function is inside another, we take the derivative of the outer function and multiply it by the derivative of the inner function.
If \( y = f(u) \) and \( u = g(x) \), then:
\( \displaystyle \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)
In Leibniz notation, :
\( \displaystyle \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)
Product Rule
When you multiply two functions together, the product rule helps you find the derivative. It states that you take the first function times the derivative of the second, and add the second function times the derivative of the first.
If \( y = u(x)v(x) \), then:
\( \displaystyle \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \)
Or in more compact notation:
\( (uv)’ = u’v + uv’ \)
Quotient Rule
If you have a function that is the ratio of two other functions, the quotient rule allows you to differentiate it. The rule says you take the denominator times the derivative of the numerator, subtract the numerator times the derivative of the denominator, and divide the whole result by the square of the denominator.
If \( y = \frac{u(x)}{v(x)} \), then:
\( \displaystyle \frac{dy}{dx} = \frac{v \frac{du}{dx} – u \frac{dv}{dx}}{v^2} \)
Or in more compact notation:
\( \left( \frac{u}{v} \right)’ = \frac{u’v – uv’}{v^2} \)
Example Differentiate \( y = \frac{x^2 \sin(x)}{e^x} \) ▶️ Answer/Explanation
|
RELATED RATES OF CHANGE
Related Rates of Change
Used when two or more variables are changing over time.
For example:
If \( x \) and \( y \) are related by a function and both are functions of \( t \), then:
\( \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \)
This is a direct application of the chain rule where time is the independent variable. Related rates are useful in geometry, physics, and motion problems.
Example Air is pumped into a spherical balloon. Volume increases at 100 cm³/s. How fast is the radius increasing when \( r = 5 \) cm? ▶️ Answer/Explanation
|
Maximum and Minimum Points & Optimisation
- Set derivative \( f'(x) = 0 \) to find stationary points.
- Use second derivative \( f”(x) \) to classify the points:
- \( f”(x) > 0 \) → local minimum
- \( f”(x) < 0 \) → local maximum
Apply in optimisation problems: minimize cost, maximize area/volume, etc.
Example Find the local maximum and minimum points of the function \( f(x) = x^3 – 6x^2 + 9x + 1 \). ▶️ Answer/Explanation
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