Maths Pure 3 · Differentiation. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
You now differentiate more than powers of x: ex, ln x, sin x, cos x, tan x and tan−1 x each have a standard derivative to learn. Angles must be in radians.
Three rules join these together. The chain rule is for a function inside a function, such as sin 3x or ln(x2 + 1). The product rule is for two functions multiplied, and the quotient rule is for one divided by another.
Some curves are not given as y = f(x). For a parametric curve, differentiate x and y with respect to the parameter and divide. For an implicit curve, differentiate every term with respect to x, and each time you differentiate a y term multiply by dy/dx.
In implicit differentiation students treat xy as a single term and write its derivative as dy/dx. It is a product, so its derivative is y + x(dy/dx). Also remember that the derivative of the constant on the right-hand side is 0.
Answer: The gradient is −3
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