# Derivatives

Derivatives are an essential part of calculus and they are used in many different fields such as engineering, business, biology, and physics. The derivative is the instantaneous rate of change of a function or simply put, the slope of a tangent line at a certain point. Derivatives are very useful because they can be used for finding minimums and maximums or optimizing a variable.

The official way of calculating a derivative is known as the limit definition of a derivative as shown in the picture above. Another way to think about this is by comparing this to the slope formula. In this case, the delta x is a very small number, so by taking the limit as delta x goes to zero allows you to find the instantaneous slope of a point.

How to take a Derivative

There are six main rules of derivatives: Power Rule, Sum Rule, Difference Rule, Quotient Rule, Product Rule, Chain Rule.

Power Rule

The power rule states that the derivative of the function x^n is simply n * xn-1. For example, the derivative of x2 would be 2x. This means that at every point we are changing at a speed of twice the current x-position.

Sum Rule

The sum rule states that the derivative of f + g is f’ + g’. The ‘ indicates the function is a derivative.

Difference Rule

The difference rule states that the derivative of f – g is f’ – g’.

Quotient Rule

The quotient rule states that the derivative of f / g is (f’ g − g’ f )/ g2

Product Rule

The product rule states that the derivative of f * g is (f * g’) + (f’ * g).

Chain Rule

The chain rule states that the derivative of f º g is (f’ º g) × g’

Applications of Derivatives

Derivatives can be used in business to maximize the profit subject to the constraints x is bound to. For example, if the profit function of an apartment complex is -8x2 + 3200x – 80000, how many of those 250 apartments should they rent in order to maximize their profit? We can solve this problem by taking a derivative to find the point where the derivative function is maximized. The maximum of the derivative function is the point where the slope is highest on the profit function which means that point allows the apartment complex to gain the most profit. It turns out that they should rent out 200 apartments to gain the most profit.

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