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How to Differentiate Fractions?

TopA function can be defined as that term in mathematics which states relationships between the constants and one or more variables. We do differentiation in order to reduce the equation with respect to a certain variable. We are familiar with basic fundamentals of differentiation. The difficulty arises in finding the Derivatives of Functions or expressions including Fractions. For example, consider the function f(x) = h(x) / g(x). Where, h(x) and g(x) are two mathematical expressions and are present in form of a fraction. Now we will see how to differentiate fractions? To resolve this we have to follow a formula given as:
d (f(x)) / dx = [(g(x) (d (h(x)) / dx) – h(x) (d (g(x)) / dx)) / g2(x)]
To understand it better let’s consider an example of a function: f(x) = (x + 1) / (x + 2). Here, let us say: h(x) = (x + 1) and g(x) = (x + 2). Applying above formula directly to differentiate the function f(x) we can write the derivative as:
d (f(x)) / dx = [((x + 2) (d ((x + 1)) / dx) – (x + 1) (d ((x + 2)) / dx)) / (x + 2)2],
or d (f(x)) / dx = [((x + 2) (1) – (x + 1) (1)) / (x2 + 4 + 4x)],
or d (f(x)) / dx = [((x + 2) (1) – (x + 1) (1)) / (x2 + 4 + 4x)],
or d (f(x)) / dx = [((x + 2 - x + 1) / (x2 + 4 + 4x)],
or d (f(x)) / dx = [3 / (x2 + 4 + 4x)].
The above answer is the called as first derivative of f(x). The higher order derivatives can also be evaluated in similar way. Remember the fraction should not result to 0, otherwise fraction would be infinity or not defined.