Product rule

In differential calculus, the product rule is a rule that helps calculate derivates that have multiplication.
formula_1
formula_2.
Steps.
Say we have the function formula_3.
The two functions being multiplied are formula_4 and formula_5.
We can set
formula_6 and
formula_7.
The rule needs us to find the derivative of both formula_8 and formula_9.
We can find formula_10 by first using the sum rule to split formula_8 into formula_12 and formula_13. After using the power rule, we have formula_14.
To find formula_15, we need to find the derivative of formula_5, which is formula_17, meaning formula_18.
Now we can substitute the values into the equation,
formula_19.
Proof.
One definition of a derivative is
formula_20, and we're trying to find the derivative of formula_21, so we can first set formula_22 to formula_21.
formula_24
We can't really do much with this so we need to manipulate the equation.
formula_25
The formula_26 part is equal to formula_27, meaning it didn't change the value of the equation. Now we can factor,
formula_28, and because formula_29 approaches formula_27, formula_31 is equal to formula_32.
formula_33, and formula_34 and formula_35 are just equal to formula_15and formula_37.
formula_2.