Differentiation (Pure 3)
Differentiation in Pure 3
- The methods are those of Pure 2: the product, quotient and chain rules, with parametric and implicit curves.
- One new standard derivative — the inverse tangent:
$$\frac{d}{dx}\tan^{-1}x = \frac{1}{1 + x^2}$$
Practice
The derivative of tan⁻¹x is 1/(1+x²). What is its value at x = 1?
1/(1 + 1²) = 1/2 = 0.5.
Practice
What is the derivative of tan⁻¹x at x = 0?
1/(1 + 0²) = 1/1 = 1.
Practice
Pure 3 differentiation still uses the product, quotient and chain rules from Pure 2.
Those rules carry over; only the inverse-tangent derivative is new.
You've got it
Key idea
- reuse the product / quotient / chain rules from Pure 2
- the new derivative: $\dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1 + x^2}$
- still handle parametric and implicit curves the same way