Differentiation (Further Pure 2)
New derivatives
- You can now differentiate the hyperbolic functions and the inverse functions $\sin^{-1}x$, $\tan^{-1}x$, $\sinh^{-1}x$.
- You can also find $\dfrac{d^2y}{dx^2}$ for implicit or parametric curves.
Practice
The derivative of sinh x is:
d/dx(sinh x) = cosh x (and d/dx(cosh x) = sinh x).
Maclaurin's series
- A Maclaurin series writes a function as a power series:
$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$
- e.g. $e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots$
Practice
The Maclaurin series of eˣ begins:
eˣ = Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + …
Practice
In a Maclaurin series, the constant term is f(0). For f(x) = eˣ, what is f(0)?
e⁰ = 1, so the constant term is 1.
You've got it
Key idea
- differentiate hyperbolic and inverse functions ($\sinh$, $\sin^{-1}$, $\tan^{-1}$…)
- a Maclaurin series expands $f$ in powers of $x$ using $f^{(n)}(0)$
- $e^x = 1 + x + \dfrac{x^2}{2!} + \cdots$