The derivative of tanx can be a tricky concept to understand for many students, but with a little bit of explanation, it can become quite clear. Here, we will delve into the world of calculus and explore the derivative of tanx.

Firstly, it is important to understand what a derivative is. At its core, a derivative is a measure of how much a function changes when its input (x in this case) changes. In calculus, we use the slope of a tangent line to measure this change.

Now, let’s look at the function tanx. Tanx is short for tangent x, which is a trigonometric function that describes the ratio of the length of the side opposite an angle in a right triangle to the length of the adjacent side. To find the derivative of tanx, we need to apply some calculus rules.

The first rule we need to use is the chain rule. The chain rule tells us how to differentiate composite functions, which are functions that are made up of two or more functions. In this case, the composite function we need to deal with is the function inside the tangent function.

The function inside the tangent function is x, so we can write this as tan(x). Using the chain rule, we need to take the derivative of the function inside the tangent function first, which is just 1. We then multiply this result by the derivative of the tangent function itself, which is sec^2(x) (where sec stands for the secant function).

Putting it all together, we get:

d/dx(tanx) = sec^2(x)

This is the derivative of tanx! So, what does this mean? Essentially, it tells us the rate at which tanx is changing for any given value of x. This can be useful in a variety of different applications, such as in physics, engineering, and economics.

In conclusion, the derivative of tanx is sec^2(x), which can be found by applying the chain rule in calculus. While it may seem daunting at first, with a little bit of practice and understanding, anyone can grasp the concept of derivatives and apply them to real-world problems.