Question

What is the derivative of $\sin \left( 3x+5 \right)$?

Answer 1

For solving this question you should know about the differentiation of trigonometric function and we can solve this by this. Here we can see that two terms are looking, so we will differentiate the outside terms first and then we differentiate the inside term. Thus we will get the answer for this.

Complete step by step answer:
According to our question, we have to calculate the derivative of $\sin \left( 3x+5 \right)$. As we know that the differentiation of trigonometric functions will always give us trigonometric values and if we differentiate to inverse trigonometric functions then it will give any other value other than trigonometric functions. And by differentiation of any variable it will reduce the power of the variable by 1 and the power will multiply by it again.
So, according to our question, we are given $y=\sin \left( 3x+5 \right)$.
So, if we consider $y=\left( f\left( x \right) \right)=\sin \left( 3x+5 \right)$,
Then if we differentiate it, then we will get as follows,
$y=\sin \left( 3x+5 \right)$
So,
$\dfrac{dy}{dx}=\dfrac{d}{dx}\left[ \sin \left( 3x+5 \right) \right]$
So, by using the chain rule here, we will get,
$\begin{aligned} & \dfrac{dy}{dx}=\cos \left( 3x+5 \right).\dfrac{d}{dx}\left( 3x+5 \right) \\\ & =\cos \left( 3x+5 \right).3 \\\ & \Rightarrow y'=3\cos \left( 3x+5 \right) \\\ & \dfrac{d}{dx}\left[ \sin \left( 3x+5 \right) \right]=3\cos \left( 3x+5 \right) \\\ \end{aligned}$
So, the differentiation of $\sin \left( 3x+5 \right)$ is equal to $3\cos \left( 3x+5 \right)$.

Note: For calculating the differentiation of any trigonometric function we have to learn all the trigonometric formulas for differentiation because this is completely based on formulas. And always we should be careful about the power of variables and specially if they are the variable whose regarding we are differentiating that.