Differentiation: standard functions, chain, product and quotient rules
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 6 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- How fast is it changing?
- The standard derivatives
- The chain rule
Read the transcript of the teaching part
Last time
Before we start: last time, in Vectors, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
How fast is it changing?
The speedometer of a car shows how quickly the distance is changing, right now. That instant rate of change is the derivative. To differentiate quickly you need a short list of standard results, and three rules for combining them.
The standard derivatives
The derivative of x to the n is n x to the n minus one. e to the x differentiates to itself. ln x differentiates to one over x. Sine goes to cosine, and cosine goes to minus sine. Tangent goes to secant squared. Cambridge expects these in radians.
The chain rule
If y is a function of u, and u is a function of x, then dy by dx equals dy by du, times du by dx. In practice: differentiate the outer function, leave the inside alone, then multiply by the derivative of the inside. The derivative of sine of five x plus three is five cosine of five x plus three.
How often it comes up
The whole chapter: Differentiation.
Chapter 14: Differentiation and integration
- Differentiation: standard functions, chain, product and quotient rules 5 min
- Tangents and normals 5 min
- Stationary points and the derivative tests 5 min
- Practical maxima and minima 5 min
- Connected rates of change and small increments 5 min
- Integration: powers of x and the reverse of differentiation 5 min
- Integrating (ax + b)ⁿ, exponentials and trigonometric functions 5 min
- Definite integrals and plane areas 5 min
- Kinematics: displacement, velocity, acceleration and graphs 5 min
Also for O Level Additional Maths 4037
Which topics come up most in O Level Additional Maths 4037 past papers, counted from every paper since 2004.