Changes

Essentially, differentiation is employed as a means to calculate the [[gradient]] or rate of change at a particular value for a given function, ''f''. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph. These are a few examples of the applications of differentiation.

Essentially, differentiation is employed as a means to calculate the [[gradient]] or rate of change at a particular value for a given function, ''f''. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph. These are a few examples of the applications of differentiation.

When defined from the first principals, the derivative of a function is the gradient of a function over <math>[x,x+h]</math>. If <math>h</math> is allowed to appoach 0 then the gradient approches the gradient at the point <math>x</math>:

When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over <math>[x,x+h]</math> as <math>h</math> tends to zero. In other words, the derivative

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

Alternative notation also commonly found is <math>y=f(x)</math> and <math>\frac{dy}{dx}=f'(x)</math>.

Alternative notation also commonly found is <math>y=f(x)</math> and <math>\frac{dy}{dx}=f'(x)</math>.