d/dx[f(x)/g(x)] (quotient rule)
To find the derivative of the function f(x)/g(x) using the quotient rule, we can follow these steps:
1
To find the derivative of the function f(x)/g(x) using the quotient rule, we can follow these steps:
1. Identify the functions f(x) and g(x) within the quotient.
2. Apply the quotient rule formula, which states:
d/dx[f(x)/g(x)] = (g(x) * f'(x) – f(x) * g'(x))/(g(x))^2
3. Calculate f'(x) and g'(x), which are the derivatives of f(x) and g(x) with respect to x, respectively.
4. Substitute these values into the quotient rule formula.
Let’s work on an example to show the application of the quotient rule.
Example: Find the derivative of the function y(x) = (3x^2 + 2x) / (x^3 + x).
Solution:
Here, f(x) = 3x^2 + 2x and g(x) = x^3 + x.
Now, let’s find f'(x) and g'(x), the derivatives of f(x) and g(x).
f'(x) = d/dx[3x^2 + 2x]
= 6x + 2
g'(x) = d/dx[x^3 + x]
= 3x^2 + 1
Next, substitute these values into the quotient rule formula:
d/dx[(3x^2 + 2x) / (x^3 + x)] = [(x^3 + x)(6x + 2) – (3x^2 + 2x)(3x^2 + 1)] / [(x^3 + x)^2]
Simplifying further:
= (6x^4 + 2x^2 + 6x^2 + 2x – 9x^4 – x) / (x^3 + x)^2
= (-3x^4 + 8x^2 + x) / (x^3 + x)^2
So, the derivative of y(x) = (3x^2 + 2x) / (x^3 + x) is (-3x^4 + 8x^2 + x) / (x^3 + x)^2.
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