Derivatives of Advanced Functions
Trigonometric Functions
In AHL 5.15, students expand their knowledge of derivatives to include more complex trigonometric functions:
- $\frac{d}{dx}(\tan x) = \sec^2 x$
- $\frac{d}{dx}(\sec x) = \sec x \tan x$
- $\frac{d}{dx}(\csc x) = -\csc x \cot x$
- $\frac{d}{dx}(\cot x) = -\csc^2 x$
To find the derivative of $y = \tan(3x)$, we use the chain rule: $\frac{dy}{dx} = \sec^2(3x) \cdot 3 = 3\sec^2(3x)$
Exponential and Logarithmic Functions
For exponential and logarithmic functions with any base $a$:
- $\frac{d}{dx}(a^x) = a^x \ln a$
- $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$
When $a = e$ (Euler's number), these simplify to the familiar forms: $\frac{d}{dx}(e^x) = e^x$ and $\frac{d}{dx}(\ln x) = \frac{1}{x}$
Inverse Trigonometric Functions
The derivatives of inverse trigonometric functions are also covered:
- $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$
Do not forget to use the chain rule on these standard derivatives!
- For example
$$ \frac{d}{dx}(a^{e^x})=a^{e^x}\ln(a)\times e^x $$ - Another Example
$$ \frac{d}{dx}(\arctan(x^2))=\frac{1}{1+x^4}(2x) $$
Indefinite Integration
Basic Indefinite Integrals
The indefinite integrals of the functions above are essentially the reverse of their derivatives:
- $\int \sec^2 x , dx = \tan x + C$
- $\int \sec x \tan x , dx = \sec x + C$
- $\int \csc x \cot x , dx = -\csc x + C$
- $\int \csc^2 x , dx = -\cot x + C$
- $\int a^x , dx = \frac{a^x}{\ln a} + C$
- $\int \frac{1}{x \ln a} , dx = \log_a |x| + C$
- $\int \frac{1}{\sqrt{1-x^2}} , dx = \arcsin x + C$
