Derivative of Quotient of Two Functions

When dealing with calculus and differentiation, one often encounters the need to take the derivative of a quotient of two functions. This operation is governed by the Quotient Rule, which is a method for finding the derivative of a function that is the ratio of two differentiable functions.

Understanding the Quotient Rule

The Quotient Rule is a formula used to find the derivative of a function that is the division of two other functions. If we have a function $h(x)$ that can be expressed as the quotient of two functions $u(x)$ and $v(x)$, where $h(x) = \frac{u(x)}{v(x)}$, then the derivative of $h(x)$ with respect to $x$ is given by:

$$ h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} $$

Here, $u'(x)$ and $v'(x)$ represent the derivatives of $u(x)$ and $v(x)$, respectively.

Important Points and Differences

Aspect	Description
Applicability	The Quotient Rule is only applicable when differentiating a ratio of two differentiable functions.
Numerator of the Derivative	The numerator consists of the derivative of the top function times the bottom function minus the top function times the derivative of the bottom function.
Denominator of the Derivative	The denominator is the square of the bottom function.
Conditions	Both functions $u(x)$ and $v(x)$ must be differentiable, and $v(x)$ must not be equal to zero.

Formulas

The Quotient Rule can be expressed as:

$$ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} $$

where $u = u(x)$ and $v = v(x)$ are differentiable functions of $x$.

Examples

Let's go through a couple of examples to illustrate the use of the Quotient Rule.

Example 1: Simple Rational Function

Consider the function $f(x) = \frac{x^2}{x + 1}$. To find $f'(x)$, we apply the Quotient Rule:

Let $u(x) = x^2$ and $v(x) = x + 1$.

Then, $u'(x) = 2x$ and $v'(x) = 1$.

Applying the Quotient Rule:

$$ f'(x) = \frac{(2x)(x + 1) - (x^2)(1)}{(x + 1)^2} = \frac{2x^2 + 2x - x^2}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2} $$

Example 2: Trigonometric Function

Consider the function $g(x) = \frac{\sin(x)}{e^x}$. To find $g'(x)$, we apply the Quotient Rule:

Let $u(x) = \sin(x)$ and $v(x) = e^x$.

Then, $u'(x) = \cos(x)$ and $v'(x) = e^x$.

Applying the Quotient Rule:

$$ g'(x) = \frac{\cos(x)e^x - \sin(x)e^x}{(e^x)^2} = \frac{e^x(\cos(x) - \sin(x))}{e^{2x}} = \frac{\cos(x) - \sin(x)}{e^x} $$

In both examples, we followed the steps of identifying $u(x)$ and $v(x)$, computing their derivatives, and then applying the Quotient Rule formula to find the derivative of the quotient.

Conclusion

The Quotient Rule is an essential tool in calculus for finding the derivative of a quotient of two functions. It is important to remember the formula and apply it correctly, ensuring that the functions involved are differentiable and that the denominator is not zero. With practice, the application of the Quotient Rule becomes a straightforward process in the differentiation of complex functions.