Question

Differentiate with respect to x.
\[sin\left( {3x + 2} \right).\]

Answer 1

Hint: To solve the given complex function, we will apply the chain rule of differentiation, because the given function is in the form of \[f\left( {g\left( x \right)} \right).\] We will start with assuming the parts of function, such as \[{\text{y }} = {\text{ }}sin{\text{ }}\left( {3x{\text{ }} + {\text{ }}2} \right)\], similarly, \[u{\text{ }} = {\text{ }}\left( {3x{\text{ }} + {\text{ }}2} \right)\]. Then on differentiating all the parts of the function with respect to the variable, and then finally connecting all the values, we will get our required answer.

Complete step by step solution:
We have been given a complex function \[sin\left( {3x + 2} \right).\] We will differentiate it using chain rule, because one function is inside of another function. Here, we can see that \[\left( {3x + 2} \right)\] is a function of sine function here.
The general formula of chain rule of differentiation is $\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}}.....eq.(1)$
Let the given function be, \[{\text{y }} = {\text{ }}sin{\text{ }}\left( {3x{\text{ }} + {\text{ }}2} \right).....eq.(2)\]
Here, we can see that \[\left( {3x{\text{ }} + {\text{ }}2} \right)\] is a function of sine function.

Then, let the inner function be, \[u{\text{ }} = {\text{ }}\left( {3x{\text{ }} + {\text{ }}2} \right)....eq.(3)\]
Now the given function has become a variable in u, we get
\[{\text{y }} = {\text{ }}sin{\text{ }}\left( u \right)\]
So, on differentiating $y = \sin u$, with respect to u, we get
$\dfrac{{dy}}{{du}} = \cos u.................(\because \dfrac{{d(\sin x)}}{{dx}} = \cos x)$
Also differentiating, \[u{\text{ }} = {\text{ }}\left( {3x{\text{ }} + {\text{ }}2} \right)\], with respect to x, we get
$\dfrac{{du}}{{dx}} = 3......................(\because \dfrac{{d(3x + 2)}}{{dx}} = 3)$

Now, on applying these values in \[eq.{\text{ (}}1),\] we get
$
  \dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}} \\
  \dfrac{{dy}}{{dx}} = \cos u \times 3 \\
$
On putting the value of u in above equation, we get
\[
  \dfrac{{dy}}{{dx}} = \cos (3x + 2) \times 3 \\
  \dfrac{{dy}}{{dx}} = 3\cos (3x + 2) \\
\]

Thus,\[\dfrac{{dy}}{{dx}} = 3\cos (3x + 2)\] is our required answer.


Note:
Students should notice that here we have used chain rule of differentiation. So, always remember in such questions, never differentiate directly, always assume some value and then differentiate with respect to the variable, using chain rule. Chain rule can be applied to many functions. It is used to compute the derivative of a composite function.