A Derivative is a measure of how the output of a specific function, which is not limited to y or f(x), changes with respect to the input. It is commonly written in the following form:

dy/dx

where,

d also known as delta, represents the change

y is the output

x is the input

Thus for a simple function such as y=5x^3, we can differentiate it in the following way:

Y = 5x^3

(d/dx)Y = (d/dx)5x^3

dy/dx = 5(d/dx)x^3

dy/dx = 5(3x^2)

dy/dx = 15x^2

However, finding the derivatives is usually never as easy as the above example. Instead there are many different methods to find the derivative of complex functions. For example, there is the quotient rule for functions with fractions; and there is the chain rule for composition functions. Consider the following function:

y = (2x+2)/(x-5)

we can split the function into two separate functions:

f(x) = 2x+2

g(x) = x-5

The quotient rule to solve this derivative is as follows:

[(f(x)/g(x)]’ = [g(x)*f(x)’ – f(x)*g(x)’]/[g(x)^2)

dy/dx = [(x-5)(2) – (2x+2)(1)]/[(x-5)^2]

dy/dx = [2x-10-2x-2]/[(x-5)^2]

dy/dx = -12/[(x-5)^2]

From this example, it can be seen that finding the derivative is not always straightforward. Thus, understanding the complexities and the multitude of different rules to differentiate a function is crucial for the study of calculus.