Product Rule

The product rule is used to difrentiate two sets of functions, for example y = 3x(x² + 4), e^x(x⁵ + 5), and x⁸ (inx + 6) If you observe these functions you will see that they contain two sets of functions and this is where the product rule is useful.
The following is the product rule and it can be proved but for this maths topic I will be contrating on how to use the rule.

product rule

Workout example

Use the product rule to differentiate y = (3x + 1)(x² + 5)

Step one will be to identify the numbers for the letters of the product rule to replace with from the function given, so we have;
u = 3x + 1 and v = x² + 5

Now we need to find dv/dx and du/dx, or we need to differentiate the u and v since this is required by the product rule.

We can now use the product rule since we know the values for the letters.

Our answer would be 9x² + 2x + 15

Example 2

Differentiate x⁵e^x

u = x⁵ and v = e^x

We can then use the product rule since we know the values.

Differentiate x²(x+inx).