A student tries to differentiate f(x)=kx3f(x)=kx^3f(x)=kx3 from first principles, where k k\,k is a constant. The student writes
f(x+h)−f(x)h=k(x3+3x2h+h3)−kx3h\dfrac{f(x+h)-f(x)}{h}=\dfrac{k(x^3+3x^2h+h^3)-kx^3}{h}hf(x+h)−f(x)=hk(x3+3x2h+h3)−kx3.
Identify the error and complete a correct first-principles derivation of f′(x)f'(x)f′(x).
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.