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The product rule (product formula). How to differentiate 2 functions that are multiplied together.

Updated on August 31, 2011


The product rule can be used to help you differentiate a function which is made up of two separate functions that are being multiplied together.

So if f(x) = g(x).h(x) then:

f`(x) = g(x)h`(x) + g`(x)h(x)

So you multiply the first function by the derivative of the second function and add this to the derivative of the first function multiplied by the second function.

Example 1

Work out f`(x) if f(x) = 5x²sin(2x)

First identify your two functions that are being multiplied together:

g(x) = 5x² and h(x) = sin(2x)

Now differentiate each of these functions:

g`(x) = 10x and h`(x) = 2cos(2x)

So all you need to do now is substitute these into the product formula for differentiation:

f`(x) = g(x)h`(x) + g`(x)h(x)

= 5x².2cos(2x) + 10x.sin(2x)

= 10x²cos(2x) + 10xsin(2x)

Example 2

Work out f`(x) if f(x) = 2cos(7x).ln(3x)

First identify your two functions that are being multiplied together:

g(x) = 2cos(7x) and h(x) = ln(3x)

Now differentiate each of these functions:

g`(x) = -14sin(7x) and h`(x) = 1/x

So all you need to do now is substitute these into the product formula for differentiation:

f`(x) = g(x)h`(x) + g`(x)h(x)

= 2cos(7x).1/x – 14sin(7x).ln(3x)

= (2/x)cos(2x) – 14sin(7x).ln(3x)

Example 3

Work out f`(x) if f(x) = cos²(5x)

This one is not as easy to identify the 2 functions. Since cos²(5x) is cos(5x)×cos(5x) then the two functions are:

g(x) = cos(5x) and h(x) = cos(5x)

Now differentiate each of these functions:

g`(x) = -5sin(5x) and h`(x) = -5sin(5x)

So all you need to do now is substitute these into the product formula for differentiation:

f`(x) = g(x)h`(x) + g`(x)h(x)

= cos(5x).-5in(5x) – 5sin(5x).cos(5x)

= -5cos(5x)sin(5x)-5cos(5x)sin(5x

= -10cos(5x)sin(5x)

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