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Lesson 5, Topic 1
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# Differentiation

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### Definition of Differentiation

Differentiation is the process of finding the derivative of a function.

It is the process of determining the rate of change of one variable with respect to another or the ratio of the rates of change of two quantities one of which is a function of the other

For example, if a particle has traveled a distance, s, at any instant of time, t, and a distance $$\scriptsize s + \delta s$$ at time $$\scriptsize t + \delta t$$, the velocity at a time is given by $$\frac {ds}{dt} \scriptsize = \lim\limits_{\delta t \to 0} \normalsize \frac{\delta s}{\delta t}$$

Note: $$\delta$$is a symbol used to represent infinitesimal changes or differences in quantities that change.

In order to find the gradient of the curve at $$\scriptsize p(x,f(x)$$, we move along the curve to $$\scriptsize p'(x + h,f(x + h ) \; where \; \scriptsize h \neq 0$$

$$\frac{\Delta y}{\Delta x} = \frac{ y_2 \; – \; y_1}{x_2 \; – \; x_1}$$

= $$\frac{ f(x + h ) \; – \; f(x)}{(x + h) \; – \; x } = \frac{ f(x + h ) \; – \; f(x)}{h}$$

Thus a function is said to be differentiable at x if –

f'(x) = $$\scriptsize \lim\limits_{h \to 0} \normalsize \frac{ f(x + h ) \; – \; f(x)}{h}$$

exists, and the derivative of f(x) with respect to x is f'(x) and is defined as:

f'(x) = $$\scriptsize \lim\limits_{h \to 0} \normalsize \frac{ f(x + h ) \; – \; f(x)}{h}$$

This is the process of finding derivatives by first principle.

There are a few different notations or symbols used to refer to the differentiation of a function, they are given as;

$$\scriptsize f'(x), y’, \normalsize \frac{dy}{dx}\scriptsize, \normalsize\frac {df(x)}{dx}\scriptsize , f_x, e.t.c$$

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