Differentiation allows us to find rates of change, for example, it allows us to find the rate of change of gradient on a curve. There are a number of simple rules which can be used to work out the derivative easily.
There are a number of ways of writing the derivative:
(1) If y = x², dy/dx = 2x
This means that if y = x², the derivative of y, with respect to x is 2x.
(2) d (x²) = 2x
This says that the derivarive of x² with respect to x is 2x.
(3) If f(x) = x², f '(x) = 2x
This says that is f(x) = x², the derivative of f(x) is 2x.
Finding the Gradient of a Curve
What is the gradient of the curve y = 2x³ when x = 3.
dy/dx = 6x²
When x = 3, dy/dx = 6×9 = 54