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Question

Question: What is the derivative of \[\sin 3x\]?...

What is the derivative of sin3x\sin 3x?

Explanation

Solution

Hint : Here, we are asked to find the derivative of sin3x\sin 3x with respect to xx. We will first let the equation be equal to yy. As we can see it consists of two functions. One of the functions is the trigonometric function whereas the other is the algebraic function. So, we will use the chain rule of differentiation to find the derivative of the given function.

Complete step-by-step answer :
Let,
y=sin3xy = \sin 3x
Now we will differentiate both the sides of the above equation with respect to xx. So, we get;
dydx=d(sin3x)dx\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\sin 3x} \right)}}{{dx}}
Since, sin3x\sin 3x contains two functions that are trigonometric and algebraic, so we will use the chain rule of differentiation.
So, according to the chain rule, as the angle in sin3x\sin 3x is 3x3x so we will first differentiate it with respect to 3x3x and then we will differentiate 3x3x with respect to xx.
dydx=d(sin3x)d(3x)×d(3x)dx\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\sin 3x} \right)}}{{d\left( {3x} \right)}} \times \dfrac{{d\left( {3x} \right)}}{{dx}}
Now, we know differentiation of sinx\sin x with respect to xx is cosx\cos x i.e., dsinxdx=cosx\dfrac{{d\sin x}}{{dx}} = \cos x.
Also, differentiation of 3x3x with respect to xx is 33. So, we get;
dydx=cos(3x)×3\Rightarrow \dfrac{{dy}}{{dx}} = \cos \left( {3x} \right) \times 3
dydx=3cos3x\Rightarrow \dfrac{{dy}}{{dx}} = 3\cos 3x
So, the correct answer is “3cos3x3\cos 3x”.

Note : For the solution we have used the chain rule of differentiation but we can also solve this using one simple shortcut or rather a formula that is;
dsin(ax+b)dx=acos(ax+b)\dfrac{{d\sin \left( {ax + b} \right)}}{{dx}} = a\cos \left( {ax + b} \right).
So, if we compare the given function that is sin3x\sin 3x with sin(ax+b)\sin \left( {ax + b} \right), then we get a=3,b=0a = 3,b = 0.
So, using the above formula we get;
dsin3xdx=3cos3x\Rightarrow \dfrac{{d\sin 3x}}{{dx}} = 3\cos 3x
Another point to note is that if in the question it is asked that what is the derivative of sin3x\sin 3x with respect to yy, then it will be zero because in that case sin3x\sin 3x will behave as constant and we know that the differentiation of constant is equal to zero.