Cambridge IGCSE Mathematics · Syllabus 0580 · Algebra and Graphs
Differentiation
What is Differentiation?
The process that turns the equation of a curve into a gradient function, written dy by dx, giving the exact gradient of the curve at every value of x rather than the single average gradient between two points. For a term a times x to the power n the derivative is a times n times x to the power n minus one, so a constant term differentiates to zero and the derivatives of a sum are added term by term. A stationary point is a point where dy by dx equals zero, and it is classified as a maximum or a minimum by an accurate sketch, by the sign of the second differential, or by inspecting the sign of the gradient on either side of it.
This definition is part of the Algebra and Graphs chapter in Cambridge IGCSE Mathematics.
Differentiation in context
A straight line has one gradient. A curve has a different gradient at every point, and until now the only way to get at it was to draw a tangent and measure. Differentiation replaces that drawing with an exact formula: feed it the equation of the curve and it hands back a second equation, \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\), whose value at any \(x\) is the gradient there. Everything else in this section is that one idea used three ways — to find a gradient, to find where the gradient is zero, and to say what kind of point that is.