Quotient Rule

The quotient rule is a formal rule for differentiating of a quotient of functions.

Let u (x) and v (x) be again differentiable functions. Then, if v (x) ≠ 0, the derivative of the quotient of these functions is calculated by the formula

To prove this formula, consider the increment of the quotient:

The derivative of the quotient is expressed as follows:

Next, using the properties of limits, we find:

Taking into account that $$ we obtain the final expression for the derivative of the quotient of two functions:

Important: The derivative of a quotient is NOT the quotient of the derivatives!

Solved Problems

Solution.

Using the quotient rule, we have

Solution.

We write the function in the form $$ and use the quotient rule.

Solution.

Let $$ $$

By the quotient rule $$ we can write

Solution.

Using the quotient rule, we have

Solution.

By the quotient rule,

At $$ the derivative is equal

Solution.

We can write the tangent function as $$. Then

As $$, $$ the derivative is given by

Solution.

As $$ we can apply the quotient rule:

Solution.

The derivative of secant can be calculated using the quotient rule: