Category: Applications of Derivatives

Related Rates

Suppose that we are given one quantity $$$x$$$ that depends on another quantity $$$y$$$. If the rate of change of the quantity $$$x$$$ with respect to the time $$$t$$$ is given, how do we find the rate of change of $$$y$$$ with respect to time? We need to do it because in real-world problems it is often easier to calculate the rate of change of $$$x$$$ than the rate of change of $$$y$$$.

Optimization Problems

Perhaps the most important application of the derivatives is solving the optimization problems.

With the help of derivatives, we can find the minimum and maximum values. That's exactly what we need in optimization problems.

Applications to Economics

Suppose that a factory produces $$$x$$$ units of some goods. Let's denote the cost of producing $$$x$$$ units of the goods by $$${C}{\left({x}\right)}$$$.

The cost function $$${C}{\left({x}\right)}$$$ is the cost of producing $$$x$$$ units of a certain product.

Tangent Line to Parametric Curves

Sometimes function is defined parametrically, but we still need to find equation of tangent line.

So, let parametric curve is defined by equations $$$x=f{{\left({t}\right)}}$$$ and $$$y=g{{\left({t}\right)}}$$$.

Tangent Line in Polar Coordinates

Let's see how to derive equation of tangent line when we are given equation of curve $$${r}={f{{\left(\theta\right)}}}$$$ in polar coordinates.

We will proceed in the same fashion as with tangent lines to parametric curves, because polar coordinates in some sense similar to parametric curves.