Derivative of tan x, sec x & tan x More

$\mathrm{The\; derivative\; of\; tan\; x,\; sec\; x\; \&\; tan\; x\; –}$The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object’s velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the “instantaneous rate of change”, the ratio of the instantaneous change in the dependent variable to that of the independent variable.

The derivative of $\mathrm{tan\; x}$

The derivative of $\tan x$ is ${\sec }^{2}x$.

To see why you’ll need to know a few results. First, you need to know that the derivative of $\sin x$ is $\cos x$. Here’s a proof of that result from first principles

Once you know this, it also implies that the derivative of $\cos x$ is $-\sin x$ (which you’ll also need later). You need to know one more thing, which is the Quotient Rule for differentiation:

Once all those pieces are in place, the differentiation goes as follows:

The derivative of sec x tan x

Suppose we have $y=f\left(x\right)\cdot g\left(x\right)$
Then, using Product Rule, $y‘=f\left(x\right)\cdot g‘\left(x\right)+f‘\left(x\right)\cdot g\left(x\right)$

In simple language, keep the first term as it is and differentiate the second term, then differentiate the first term and keep the second term as it is or vice-versa.

Derivative of tanx^-1

Derivative of tanx^2

In differential calculus, we use the Chain Rule when we have a composite function. It states that The derivative will be equal to the derivative of the outside function with respect to the inside, times the derivative of the inside function. Let’s see what that looks like mathematically

Chain Rule

$f‘\left(g\left(x\right)\right)\cdot g‘\left(x\right)$

Let’s say we have the composite function $\sin \left(5x\right)$. We know:

$f\left(x\right)=\sin x\Rightarrow f‘\left(x\right)=\cos x$

$g\left(x\right)=5x\Rightarrow g‘\left(x\right)=5$

So the derivative will be equal to

$\cos \left(5x\right)\cdot 5$

$=5\cos \left(5x\right)$

We just have to find our two functions, find their derivatives and input into the Chain Rule expression.

using the chain rule for derivative of tanx^2

Use the chain rule…