What is a derivative and how are they used?

A derivative is a function that tells us the gradient of a curve at any point. Say you have a function like f(x)=x³+x² and you want to know when the function is stationary, i.e. has a gradient of zero. We first take the derivative of f(x). This derivative function which comes to be: f'(x)=3x²+2x can be used to find the values of x for when the gradient is zero by setting this derivative to be equal to zero and then solving.In conclusion, a derivative function can be thought of as a gradient function, and it used to find specific values of x for which the gradient of the initial function is equal to some constant that is required, most commonly zero