Question

Find the derivative of \[\tan (3x)\] ?

Answer 1

Hint: Derivative of trigonometric functions are already defined but for the basic identity, if you find any changes in the angle given the you have to first expand or convert the given identity into basic then you can further differentiate the function, for example integral of “sinx” is “cosx”, is used as.

Complete step by step solution:
The given question is \[\tan (3x)\]. After implying differentiation sign we get: \[\dfrac{d}{{dx}}\tan (3x)\]
Here we have to use a substitution method to solve the question, in substitution method we have to assume a variable say “y” and “u” such that “tan(3x)” is equal to the variable to “y” and “3x” is equal to “u”.
Now on further solving we get,
\[ y = \tan 3x \\
\Rightarrow u = 3x \\
\Rightarrow \dfrac{{du}}{{dx}} = 3 \\ \]
Now substituting these values we get:
\[\Rightarrow y = \tan u \\
\Rightarrow \dfrac{{dy}}{{du}} = {\sec ^2}u\left( {\dfrac{d}{{dx}}(\tan x) = {{\sec }^2}x} \right) \\
\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}} \\
\Rightarrow \dfrac{{dy}}{{dx}} = ({\sec ^2}u) \times 3 \\
\therefore\dfrac{{dy}}{{dx}}= 3{\sec ^2}3x \]
This is our required differentiation.

Note: This is the easiest way to solve this question, but if you find any difficulty then you can try another method that is, convert the equation in term of “sinx and cosx” then solve for that and the same result you will be getting after you again change the “sinx and cosx” term in “tanx” term. Changing it into “sinx and cosx” is not a good option here because the differentiation of “tan3x” is already simple to solve and get the result. You can check the answer by solving from bottom to top; just you have to integrate the last equation until you get the first equation and accordingly integrate the term from bottom.