Question

How do you differentiate $f(x)=\cot (3x)$ using the chain rule?

Answer 1

The given expression is to be differentiated using the chain rule. We will first differentiate the cotangent function which is equal to $-{{\csc }^{2}}x$. Then, we will proceed to differentiate the angle of cotangent function which is $3x$. When we differentiate $3x$, we get 3. Hence, we have the differentiation of the given expression.

Complete step-by-step solution:
According to the given question, we are provided with an expression which we have to differentiate using the chain rule.
Chain rule as we know is a rule that tells the sequence in which an expression is differentiated. Using chain rule, we go from the outermost function to the innermost function, differentiating them step by step.
For example – if we have an expression suppose, $y={{e}^{\sin x}}$
Then we will first differentiate the exponential function and then we will differentiate the power raised in the exponential function. That is,
$y'={{e}^{\sin x}}.\cos x$
The given function we have is,
$f(x)=\cot (3x)$----(1)
We will be using the chain rule to differentiate the above function. We have,
$f'(x)=-{{\csc }^{2}}(3x)$----(2)
As we know that derivative of cotangent function is $-{{\csc }^{2}}x$, that is, $\dfrac{d}{dx}\cot x=-{{\csc }^{2}}x$
Next, we will differentiate the $3x$, continuing in equation (2), we get,
$f'(x)=-{{\csc }^{2}}(3x).3$
As we know that derivative of $3x$ is 3, that is, $\dfrac{d}{dx}(3x)=3$
Therefore, the differentiation of the given function is $f'(x)=-3{{\csc }^{2}}(3x)$.

Note: The chain rule is very effective when we are given with a composite function. Since the chain rule tells the sequence in which the function is to be differentiated. A composite function is a one which has a function within a function.
For example- $\cot (3x)$ is a composite function.