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SS3: MATHEMATICS - 2ND TERM

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  1. Matrices I | Week 1
    6 Topics
  2. Matrices II | Week 2
    1 Topic
    |
    1 Quiz
  3. Commercial Arithmetic | Week 3
    7 Topics
    |
    1 Quiz
  4. Coordinate Geometry | Week 4
    8 Topics
    |
    1 Quiz
  5. Differentiation of Algebraic Expressions | Week 5 & 6
    7 Topics
  6. Application of Differentiation | Week 7
    4 Topics
    |
    1 Quiz
  7. Integration | Week 8
    8 Topics
    |
    1 Quiz



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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.

differentiation

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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