quotient rule example
The quotient rule is used to find the derivative of a function that is the quotient of two other functions. Mathematically, it can be stated as:
If f(x) = g(x) / h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Here's an example to illustrate the quotient rule:
Let's find the derivative of f(x) = x^2 / (2x + 1).
Using the quotient rule, we have:
f'(x) = [(2x + 1)(2x) - (x^2)(2)] / (2x + 1)^2
Simplifying the numerator:
f'(x) = (4x^2 + 2x - 2x^2) / (2x + 1)^2
Combining like terms:
f'(x) = (2x^2 + 2x) / (2x + 1)^2
Simplifying further is not possible in this case. So, that's the derivative of f(x) using the quotient rule.
Sure! The quotient rule is a formula used to find the derivative (rate of change) of a quotient of two functions. The formula for the quotient rule is:
\[\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\]
Let's use an example to understand how to apply the quotient rule.
Example: Find the derivative of the function \[y = \frac{3x^2 - 4}{x+2}\]
Step 1: Identify the functions f(x) and g(x). In this example, f(x) is the numerator \[(3x^2 - 4)\] and g(x) is the denominator \[(x+2)\].
Step 2: Determine the derivatives of f(x) and g(x). The derivative of f(x) is denoted as f'(x) and the derivative of g(x) is denoted as g'(x).
The derivative of f(x) = \[3 * 2x^{2-1} - 0 = 6x\] (using the power rule).
The derivative of g(x) = \[1 * 1 = 1\] (the derivative of x is 1).
Step 3: Apply the quotient rule formula.
Using the quotient rule, the derivative of the given function y = \[\frac{3x^2 - 4}{x+2}\] is:
\[\frac{d}{dx}\left(\frac{3x^2 - 4}{x+2}\right) = \frac{(6x)(x+2) - (3x^2 - 4)(1)}{(x+2)^2}\]
Step 4: Simplify the expression.
Expanding and simplifying the numerator, we get:
\[(6x)(x+2)-(3x^2-4)(1) = 6x^2 + 12x - (3x^2 - 4) = 6x^2 + 12x - 3x^2 + 4 = 3x^2 + 12x + 4\]
So, the derivative of y = \[\frac{3x^2 - 4}{x+2}\] is:
\[\frac{d}{dx}\left(\frac{3x^2 - 4}{x+2}\right) = \frac{3x^2 + 12x + 4}{(x+2)^2}\]
That's it! We have successfully found the derivative of the given function using the quotient rule.
The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. It can be stated as follows:
If you have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) with respect to x can be found using the quotient rule as:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's walk through an example to better understand the quotient rule:
Example: Find the derivative of the function f(x) = (3x^2 + 2x) / (2x^3 - x)
Step 1: Identify the functions g(x) and h(x) in the quotient. In this case, g(x) = 3x^2 + 2x and h(x) = 2x^3 - x.
Step 2: Calculate the derivatives of g(x) and h(x) individually. g'(x) represents the derivative of g(x) and h'(x) represents the derivative of h(x). In this example, g'(x) = 6x + 2 and h'(x) = 6x^2 - 1.
Step 3: Apply the quotient rule formula:
f'(x) = (g'(x)*h(x) - g(x)*h'(x)) / (h(x))^2
Substitute the values we found for g(x), g'(x), h(x), and h'(x) into the formula:
f'(x) = ((6x + 2)*(2x^3 - x) - (3x^2 + 2x)*(6x^2 - 1)) / (2x^3 - x)^2
Step 4: Simplify the expression if possible. You can expand and combine like terms to make it easier to read.
f'(x) = (12x^4 - 6x^2 + 4x^2 - 2x - 18x^4 + 9x - 12x^3 + 6x^2) / (2x^3 - x)^2
f'(x) = (-6x^4 - 6x^3 + 9x - 2x) / (2x^3 - x)^2
Step 5: Continue to simplify the expression if possible. In this case, we can combine the terms on the numerator and factor out a common factor of x in order to simplify further.
f'(x) = -6x^3 (x + 1) / (2x^3 - x)^2
And that's the derivative of the function f(x) using the quotient rule.