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How do you take the derivative of $\tan \left( \dfrac{\pi x}{2} \right)?$

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Last updated date: 20th Jun 2024
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Answer
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Hint: The quotient rule is method of determining the differentiation of a function in which the ratio of two function is differentiable by applying variant rule,
$\tan \left( x \right)=\dfrac{\sin x}{\cos x}\to \tan \left( x \right)$
$=\dfrac{{{\cos }^{2}}\left( x \right)+{{\sin }^{2}}\left( x \right)}{{{\cos }^{2}}\left( x \right)}$
The quotient rule is applicable when in the given problem one function is divided to another function.

Complete step by step solution:We know that,
For every function $h(x)$ that can be written as $f(g(x)),h'(x)=f'(g(x)).g'(x)$
In this case, we have
$f(x)=\tan (x),g(x)=\dfrac{\pi .x}{2}$
The derivative for $g(x)$ can be easily compute,
$g'(x)=\dfrac{\pi }{2}$
Since, the leading coefficient is the derivative for a first degree polynomial.
The derivative for $f(x)$ you can easily check it from the reference table or try to remember it, or we can use quotient rules.
$\tan \left( x \right)=\dfrac{\sin \left( x \right)}{\cos \left( x \right)}\to \tan \left( x \right)$
$=\dfrac{{{\cos }^{2}}\left( x \right)+{{\sin }^{2}}\left( x \right)}{{{\cos }^{2}}\left( x \right)}$
Using identity of ${{\cos }^{2}}\left( x \right)+{{\sin }^{2}}\left( x \right)=1$we get that,
$\tan '\left( x \right)=\dfrac{1}{{{\cos }^{2}}\left( x \right)}$
$\therefore \tan '\left( x \right)={{\sec }^{2}}\left( x \right)$
So, the derivative of
$\tan \left( \dfrac{\pi .x}{2} \right)=\dfrac{\pi }{2}.{{\sec }^{2}}\left( \dfrac{\pi .x}{2} \right)$

Additional Information:
The term derivative is the rate of change of function with respect to a variable. It is fundamental to the solution of problems in differential equations.
For example, the derivative $\tan '\left( x \right)=\dfrac{1}{{{\cos }^{2}}\left( x \right)}$
${{\sec }^{2}}\left( x \right)$ another in differentiation is rate of change of function or it is a process for finding derivatives. There is another basic rule, also known as chain rule. They provide a way to composite function.

Note:
Before solving any type first we have to check the type of problem, do not start if directly, it gets incorrect. After solving, check all the possibilities. Where we have taken all the derivative where it is required. If the derivative is incorrect in any steps the whole problem gets wrong. There are various types of rules which we have to apply in several various problems. Sometimes the problem cures in various forms such as fraction or in others in that we have to use quotient rule any many other rules. These are some key points which we have to remember while solving derivative problems.