Question

How do you find the derivative of $y = \cos 3x$?

Answer 1

Here we will find the derivative of $y$ with respect to $x$ by using the differentiation method. Here we will use the chain rule method and formula of differentiation of cosine function to get the required answer. Differentiation is used to calculate the instantaneous rate of change in the function given because of one of its variables.

Complete step by step solution:
Here we will find the derivative of $y = \cos 3x$ with respect to $x$.
Now differentiating both sides of the give equation, with respect to $x$, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\cos 3x} \right)}}{{dx}}$
We will use the chain rule and first find the derivative of the cosine function.
Using the formula $\dfrac{d}{{dx}}\cos x = - \sin x$, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = - \sin 3x\dfrac{{d\left( {3x} \right)}}{{dx}}$
Now, we will find the derivation of $3x$ with respect to $x$.
Using the formula of differentiation $\dfrac{{d\left( x \right)}}{{dx}} = 1$, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = - \sin 3x \times 3$
Multiplying the terms, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = - 3\sin 3x$

Therefore, the derivative of $y = \cos 3x$ is $- 3\sin 3x$.

Note:
Differentiation is done with respect to an independent variable of the function. Some real-life application of differentiation is the rate of change of velocity with respect to time. It is also used to find the tangent and normal curve as also to calculate the highest and lowest point of the curve in a graph. Differentiation of trigonometric function is a very vast topic where differentiation of different trigonometric values has different formulas. Differentiation of a constant value is always zero because differentiation measures the rate of change of a function with respect to the variable but the constants don’t change their derivative so their differentiation is zero.