The volume of a sphere is increasing at a constant rate of 2 cm3 ^3\,3s−1^{-1}−1.
The volume of a sphere of radius r r\,r cm is 43πr3\displaystyle \frac{4}{3}\pi r^334πr3 cm3^33, and its surface area is 4πr2 4\pi r^2\,4πr2 cm2^22.
Show that the rate of increase of the radius when r=2r = 2r=2 is aπ\displaystyle \frac{a}{\pi}πa cm s−1^{-1}−1, where a a\,a is a constant to be found.
Find the rate at which the surface area is increasing when r=2r = 2r=2.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.