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# Examples of Differentiation

Here are a few examples of differentiation for various types of equations – obtained by by applying the rules (product rule, quotient rule, etc) relevant to each type of equation.

Find the derivatives for the following equations:

1. x3 – 2x2 + 2
2. y = 5x4 + 10x2 + 2
3. y = (5x + 2)(2x2 + 2)
4. y = x-3(5 +2x)
5. S = with respect to r (written as wrt r)

## Now let us solve these examples of differentiation:

Example 1:  x3 – 2x2 + 2

=

= 3x2 – 2 x 2x + 0

= 3x2 – 4x

Example 2: y = 5x4 + 10x2 + 2

=

= 4 x 5x3 + 2 x 10x + 0

= 20x3 + 20x

Example 3:  y = (5x + 2)(2x2 + 3)

We can express this function as a product of two equations, and apply the product rule for differentiation.

f(x) = (5x + 2)                g(x) = (2x2 + 3)f'(x) = 5

g'(x) = 4x

=  f'(x) g(x)  +  g'(x) f(x)

= 5(2x2 + 3) + 4x(5x + 2)

= 10x2 + 15 + 20x2 + 8x

= 30x2 + 8x + 15

Example 4: y = x-3(5 +2x)

Again we can express this function as a product of two equations, and apply the product rule for differentiation.

f(x) = x-3                         g(x) = (5 + 2x)f'(x) = -3x-4

g'(x) = 2f(x)g(x)

=  f'(x)g(x) + g'(x)f(x)

= (-3x-4)(5 + 2x) + 2x-3

=

=

=

We could have solved the above equation using the Quotient Rule also, as illustrated below –

y  =    =

y’  =

=

=

=       (divide numerator and denominator by x2)

=

This is the same answer we got above by using the product rule.

Example 5: S =

Differentiating with respect to r (written as wrt r) –

=   +

=   Remember and h are constants here, only r is the variable that can be differentiated.

Learn more about first principles of derivatives, and finding derivatives using formula