Question

What is the derivative of $\tan {x^3}$ ?

Answer 1

Hint : In the given problem, we are required to differentiate $\tan {x^3}$ with respect to x. Since, $\tan {x^3}$ is a composite function, we will have to apply the chain rule of differentiation in the process of differentiating $\tan {x^3}$ . So, differentiation of $\tan {x^3}$ with respect to x will be done layer by layer using the chain rule of differentiation. Also the derivative of $\tan x$ with respect to $x$ must be remembered.

Complete step-by-step answer :
So, we have, $\dfrac{d}{{dx}}\left( {\tan {x^3}} \right)$
Now, Let us assume $u = {x^3}$. So substituting ${x^3}$ as $u$, we get,
$\Rightarrow \dfrac{d}{{dx}}\left( {\tan u} \right)$
Now, we know that the derivative of tangent function $\tan x$ with respect to x is ${\sec ^2}x$. So, we get,
$\Rightarrow {\sec ^2}u\left( {\dfrac{{du}}{{dx}}} \right)$
Now, putting back $u$as ${x^3}$, we get,
$\Rightarrow {\sec ^2}\left( {{x^3}} \right)\left( {\dfrac{{d\left( {{x^3}} \right)}}{{dx}}} \right)$
Now, we know the power rule of differentiation. According to the power rule of differentiation, the derivative of ${x^n}$ with respect to x is $n{x^{n - 1}}$. So, the derivative of ${x^3}$ with respect to x is $3{x^2}$.
Hence, we get,
$\Rightarrow {\sec ^2}\left( {{x^3}} \right) \times \left( {3{x^2}} \right)$
Simplifying the product of two terms, we get,
$\Rightarrow 3{x^2}{\sec ^2}\left( {{x^3}} \right)$
So, the derivative of $\tan {x^3}$ with respect to $x$ is $3{x^2}{\sec ^2}\left( {{x^3}} \right)$.
So, the correct answer is “ $3{x^2}{\sec ^2}\left( {{x^3}} \right)$”.

Note : In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. Remember the derivative of a constant is always zero.