Product And Quotient Rule

Product And Quotient

The product rule and quotient rule are two important rules in calculus that allow us to find the derivative of a function that is the product or quotient of two other functions.

The product rule states that the derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. This can be written mathematically as follows:

$$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{dg}{dx} + g(x) \cdot \frac{df}{dx}$$

Here is an example of the product rule in action:

Suppose we have two functions f(x) and g(x) defined as follows:

$$f(x) = x^2 + 1$$

$$g(x) = x^3 + 2x$$

To find the derivative of the product f(x)g(x), we use the product rule as follows:

$$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{dg}{dx} + g(x) \cdot \frac{df}{dx}$$

The derivative of f(x) with respect to x is simply 2x, and the derivative of g(x) with respect to x is 3x^2 + 2. Substituting these into the equation above, we get:

$$\frac{d}{dx}[f(x) \cdot g(x)] = (x^2 + 1) \cdot (3x^2 + 2) + (x^3 + 2x) \cdot 2x$$

Evaluating this expression, we get:

$$\frac{d}{dx}[f(x) \cdot g(x)] = 3x^4 + 6x^2 + 2x^3 + 4x + 2x^2 + 2x$$

This can be simplified to:

$$\frac{d}{dx}[f(x) \cdot g(x)] = 3x^4 + 8x^3 + 8x^2 + 6x$$

Therefore, the derivative of the product of f(x) and g(x) is 3x^4 + 8x^3 + 8x^2 + 6x.

The quotient rule states that the derivative of the quotient of two functions is equal to the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This can be written mathematically as follows:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot \frac{df}{dx} - f(x) \cdot \frac{dg}{dx}}{g(x)^2}$$

Here is an example of the quotient rule in action:

Suppose we have two functions f(x) and g(x) defined as follows:

$$f(x) = x^2 + 1$$

$$g(x) = x^3 + 2x$$

To find the derivative of the quotient f(x)/g(x), we use the quotient rule as follows:

Suppose we have two functions f(x) and g(x) defined as follows:

$$f(x) = x^2 + 1$$

$$g(x) = x^3 + 2x$$

To find the derivative of the quotient f(x)/g(x), we use the quotient rule as follows:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot \frac{df}{dx} - f(x) \cdot \frac{dg}{dx}}{g(x)^2}$$

The derivative of f(x) with respect to x is simply 2x, and the derivative of g(x) with respect to x is 3x^2 + 2. Substituting these into the equation above, we get:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{(x^3 + 2x) \cdot 2x - (x^2 + 1) \cdot (3x^2 + 2)}{(x^3 + 2x)^2}$$

Evaluating this expression, we get:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{2x^4 + 4x^2 - 3x^4 - 6x^2 - 2x^2 - 2}{x^6 + 4x^4 + 4x^2}$$

This can be simplified to:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{-x^4 - 2x^2 + 2x^2 + 2}{x^6 + 4x^4 + 4x^2}$$

And further simplified to:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{-x^4 + 2x^2 + 2}{x^6 + 4x^4 + 4x^2}$$

Therefore, the derivative of the quotient of f(x) and g(x) is -x^4 + 2x^2 + 2.