The quotient rule is an essential tool for differentiating functions that are expressed as one function divided by another. In order to better understand this formula, let's take a look at its structure and practice using it through some examples.
A quotient function is a type of mathematical expression where one function is divided by another. This can be written as f(x) = g(x)/h(x). In order to differentiate this type of function, the quotient rule comes into play.
The equation for the quotient rule is as follows:
dy/dx = (v du/dx - u dv/dx)/v^2 if y = u/v
In function notation, it can also be written as:
If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x))^2
Let's take a look at some problems and see how the quotient rule can be applied.
To start, we can break down the equation and identify the necessary values for the quotient rule:
Since the function is written in function notation and involves a fraction, it is evident that the quotient rule formula in function notation is needed:
By substituting these values into the formula, we can solve for dy/dx.
It's important to be able to use the quotient rule in function notation, as it may appear in exam questions. Let's review the formula in terms of function notation:
If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x))^2
Let's apply the quotient rule to solve a problem and find a specific value.
Problem: Find the value of f'(2) for the point (2, 1/3) on the curve where f(x) = (x^2 + 2x)/3x.
First, we can identify the necessary elements of the formula and calculate them:
v = 3xdu/dx = 2x + 2dv/dx = 3
Next, we can substitute these values into the formula to solve for dy/dx.
dy/dx = ((3x)(2x + 2) - (x^2 + 2x)(3))/(3x)^2
Finally, we can substitute x = 2, since that is the given x-coordinate, to find f'(2).
dy/dx = ((3(2))(4 + 2) - (4 + 4)(3))/(3(2))^2= (18 - 24)/36= -1/6
Therefore, f'(2) = -1/6.
