Question

How do you find the derivative of $y=\cos \left( 4{{x}^{3}} \right)$?

Answer 1

To get the derivative of $y=\cos \left( 4{{x}^{3}} \right)$ with respect to $x$ . Firstly, suppose $\theta =4{{x}^{3}}$ and get the derivative of $\theta$ with respect to $x$ . Now we can write $y=\cos \left( 4{{x}^{3}} \right)$ as $y=\cos \left( \theta \right)$ and after that get the derivative with respect to $\theta$ . After combining both the derivative as $\dfrac{dy}{dx}=\dfrac{dy}{d\theta }\times \dfrac{d\theta }{dx}$ we can get the derivative of $y=\cos \left( 4{{x}^{3}} \right)$ with respect to $x$ .

Complete step by step solution:
$y=\cos \left( 4{{x}^{3}} \right)$ is the given equation in the question.
Since, we are not able to derive the given equation directly. So, we can consider $4{{x}^{3}}$as:
$\theta =4{{x}^{3}}$ … $\left( i \right)$
Now, we can derive the above equation $\left( i \right)$ with respect to $x$ as:
$\Rightarrow \dfrac{d\theta }{dx}=\dfrac{d\left( 4{{x}^{3}} \right)}{dx}$
Since, numbers are constant in any derivative, so we cannot derive $4$. For ${{x}^{3}}$ , the derivative is $3{{x}^{2}}$ .
So, the derivation can be written as:
$\Rightarrow \dfrac{d\theta }{dx}=4\times 3{{x}^{2}}$
After simplifying the derivation, it would be as:
$\Rightarrow \dfrac{d\theta }{dx}=12{{x}^{2}}$ … $\left( ii \right)$
After using equation $\left( i \right)$, we can write the given equation $y=\cos \left( 4{{x}^{3}} \right)$ as-
$\Rightarrow y=\cos \left( \theta \right)$
Since, The derivative of $\cos \left( \theta \right)$ with respect to $\theta$ is $-\sin \left( \theta \right)$ . After derivative the above equation with respect to $\theta$ , we will have:
$\Rightarrow \dfrac{dy}{d\theta }=-\sin \left( \theta \right)$
Now, with the use of equation $\left( i \right)$, we can write the above derivative in term of $x$ as-
$\Rightarrow \dfrac{dy}{d\theta }=-\sin \left( 4{{x}^{3}} \right)$ … $\left( iii \right)$
For getting the derivative of the given equation $y=\cos \left( 4{{x}^{3}} \right)$ with respect to $x$, we will use the following formula:
$\dfrac{dy}{dx}=\dfrac{dy}{d\theta }\times \dfrac{d\theta }{dx}$
Using equation $\left( ii \right)$ and $\left( iii \right)$ in above formula, we will get:
$\Rightarrow \dfrac{dy}{dx}=-\sin \left( 4{{x}^{3}} \right)\times 12{{x}^{2}}$
After simplification, we can write the above equation as:
$\Rightarrow \dfrac{dy}{dx}=-12{{x}^{2}}\sin \left( 4{{x}^{3}} \right)$
Hence, the derivative of the given equation $y=\cos \left( 4{{x}^{3}} \right)$ is $-12{{x}^{2}}\sin \left( 4{{x}^{3}} \right)$ .

Note:
Here we can check whether the derivative of the given equation is correct or not in the following way-
From the solution, we have:
$\dfrac{dy}{dx}=-12{{x}^{2}}\sin \left( 4{{x}^{3}} \right)$
We can write it as-
$\Rightarrow dy=-12{{x}^{2}}\sin \left( 4{{x}^{3}} \right)dx$
After applying the symbol of integration both sides:
$\Rightarrow \int{{}}dy=\int{\left[ -12{{x}^{2}}\sin \left( 4{{x}^{3}} \right) \right]}dx$
After integrating the above equation, we will get:
$\begin{aligned} & \Rightarrow y=-\dfrac{12{{x}^{2}}}{3{{x}^{2}}}\left[ -\cos \left( 4{{x}^{3}} \right) \right] \\\ & \Rightarrow y=-4\left[ -\cos \left( 4{{x}^{3}} \right) \right] \\\ & \Rightarrow y=4\cos \left( 4{{x}^{3}} \right) \\\ \end{aligned}$
Now, we got the given equation of the question from the integration of the solution. Hence, the solution is correct.