Question

1-22: Differentiate.

3. \(y = {x^2} + \cot x\)

Short answer

Therequired value is \(y' = 2x - {\csc ^2}x\).

Step-by-step solution

Write the formula of the derivatives of trigonometric functions and the power rule

\(\frac{d}{{dx}}\left( {\cot } \right) = {\csc ^2}x\)

The Power Rule: \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\)where \(n\) is a positive integer

Find the differentiation of the function

Consider the function \(y = {x^2} + \cot x\). Differentiate the function w.r.t \(x\) by using the derivatives of trigonometric functions and the power rule.

\(\begin{aligned}\frac{{d\left( y \right)}}{{dx}} &= \frac{{d\left( {{x^2} + \cot x} \right)}}{{dx}}\\ &= \frac{d}{{dx}}\left( {{x^2}} \right) + \frac{d}{{dx}}\left( {\cot x} \right)\\ &= 2{x^{2 - 1}} + \left( { - {{\csc }^2}x} \right)\\ &= 2x - {\csc ^2}x\end{aligned}\)

Thus, the derivative of the function \(y = {x^2} + \cot x\) is \(y' = 2x - {\csc ^2}x\).