Simple Algebraic Differentiation

Differentiation From First Principles

In exams it is sometimes required that Differentiation be carried out from first principles.

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The Differential Coefficient (gradient Function)

is known as the Gradient

Function and represents the Derivative of y with respect of x. It is also known as the Differential Coefficient.

In the simplest case:-

Example 1:

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Example 2:

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It is not possible to differentiate the above equation term by term. It must either be multiplied out or treated as a product of two variables. ( See "Product Rule") The choice of which to use depends on which is the best solution for a particular equation.

Then this must be multiplied out to give:-

And So:-

Example 3:

Find the gradient of the curve at x=1

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Example 4:

If the distance traveled by a particle in time t is given by:-

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Where u and a are constants show that the velocity at time t is

Note. Velocity is the rate at which a particle moves in a straight line from a fixed point and is thus

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Note. For those taking Applied Mathematics, this is one of the equations of Motion for a Particle under constant acceleration.

Example 5:

Differentiate with respect to x

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And find the slope of the Tangent at

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Differentiating

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When x=2

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When x=1.4

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Note the tangent to the graph at x=1.4 is parallel to the x axis.

The Differentiation 0f A Product Of Two Functions Of X

It is obvious, that by taking two simple factors such as 5 X 8 that the total increase in the product is Not obtained by multiplying together the increases of the separate factors and therefore the Differential Coefficient is not equal to the product of the d.c's of its factors.

Proof Of The Above Equation:

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Where u and v are both functions of x. Thus when x increases to u and v will also change to:-

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Their product y will therefore become

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In the limit as tend to zero the above equation becomes:-

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Example 6:

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Example 7:

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Example 8:

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Example 9:

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Example 10:

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Multiplying up

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The Differentiation Of A Product Of Any Number Of Functions Of X

The rule for finding the differential coefficient of a product of two functions of x can be extended to apply to the product of any finite numbers of functions of x

Where u, v, w are all functions of x, then regarding this as the product of the two factors un and w:-

And similarly for any finite number of factors.

Note An important result follows from the above rules. The differential coefficient of with respect to x can be considered to be the product of two factors each of x and hence is given by:-

Similarly, if n is any interger, by taking the product of n factors each of y

The Differentiation Of A Quotient Of Two Functions Of X

The following is the proof from first principles of the above equation.

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As with previous proofs from first principles

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In the limit when tends to zero the so will

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Example 11:

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Example 12:

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Example 13:

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