Substituting values before differentiating

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

1. 01

   The table gives values of two differentiable functions $f$ and $g$ and of their derivatives. Let $h(x) = 3f(x) - 2g(x) + x^{3}$. Find $h'(2)$.

   ·Differentiation rules·Introductory

2. 02

   A circular ripple on a pond expands so that its radius grows at a constant 2 cm/s. How fast is the area inside the ripple increasing at the instant the radius is 4 cm?

   ·Related rates·Introductory

3. 03

   Evaluate the limit by recognising it as a derivative. $\displaystyle \lim_{x \to \pi/6} \frac{\sin x - \tfrac{1}{2}}{x - \tfrac{\pi}{6}}$

   ·Exponential and trigonometric functions·Standard

4. 04

   The point $(2, 1)$ lies on the curve below. Find $\dfrac{dy}{dx}$ at that point. $\displaystyle x^{3} + \sin(\pi y) + y = 9$

   ·Implicit differentiation and inverse trigonometric functions·Standard

5. 05

   The curve below passes through exactly one point with $y = 0$. Find $\dfrac{dy}{dx}$ at that point. $\displaystyle x\,e^{y} + 2x^{2}\sin y = 1$

   ·Implicit differentiation and inverse trigonometric functions·Standard

6. 06

   The point $(-1, 2)$ lies on the curve below. Find $\dfrac{d^{2}y}{dx^{2}}$ at that point. $\displaystyle x^{3} + y^{3} = 7$

   ·Implicit differentiation and inverse trigonometric functions·Challenging