Simplified Quotient Rule | How to Find the Derivative of (u/v) with Respect to x

dy/dx u/v

To find the derivative of the function (u/v) with respect to x, using the quotient rule of differentiation, we follow these steps:

Step 1: Identify the functions u(x) and v(x)

To find the derivative of the function (u/v) with respect to x, using the quotient rule of differentiation, we follow these steps:

Step 1: Identify the functions u(x) and v(x).
– Let u(x) be a differentiable function of x.
– Let v(x) be another differentiable function of x.

Step 2: Apply the quotient rule.
The quotient rule states that if we have a function f(x) = u(x) / v(x), then the derivative of f(x), denoted as f'(x) or df(x)/dx, is given by:

f'(x) = (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2

In our case, u(x) is represented as u, and v(x) is represented as v. Therefore, applying the quotient rule, we get:

dy/dx = (v * du/dx – u * dv/dx) / v^2

Note: The d/dx here represents the derivative operator, and du/dx and dv/dx represent the derivatives of u(x) and v(x) with respect to x, respectively.

Now, let’s simplify the expression further if needed or if more information is provided.

Definitions:

1. Derivative: The derivative of a function represents the rate at which the function changes with respect to its input variable or independent variable. It measures the instantaneous rate of change of the function.

2. Quotient Rule: The quotient rule is a formula used for finding the derivative of a function that is expressed as the ratio of two differentiable functions. It states that if f(x) = u(x) / v(x), then the derivative of f(x) is given by (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2.

Note: It is important to note that if you provide specific functions u(x) and v(x), I can show you a step-by-step solution for finding the derivative accordingly.

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