Develop a rule for the following:
Dx[f(x)g(x)/h(x)]
To develop a rule for finding the derivative of the expression Dx[f(x)g(x)/h(x)], we can use the quotient rule. The quotient rule states that if you have a function f(x) divided by g(x), the derivative of this expression can be found as follows:
If y(x) = f(x)/g(x), then
d/dx [f(x)/g(x)] = [g(x)*f'(x) - f(x)*g'(x)] / [g(x)]^2
Now, let's apply the quotient rule to the given expression Dx[f(x)g(x)/h(x)]:
We can rewrite Dx[f(x)g(x)/h(x)] as (1/h(x)) * [f(x)g(x)].
Let y(x) = f(x)g(x) and z(x) = h(x).
Now, using the quotient rule, we have:
d/dx [(1/h(x)) * y(x)] = [(h(x)*(d/dx[y(x)]) - (d/dx[h(x)]) * y(x)] / [h(x)]^2
Replacing y(x) with f(x)g(x), we get:
= [h(x)*(d/dx[f(x)g(x)]) - (d/dx[h(x)]) * f(x)g(x)] / [h(x)]^2
Therefore, the rule for finding the derivative of Dx[f(x)g(x)/h(x)] is to apply the quotient rule, as shown above.
Keep in mind that you'll need the derivatives of f(x), g(x), and h(x) to fully evaluate the expression. You can find these derivatives using other rules, such as the product rule and the chain rule, depending on the complexity of the functions involved.