Differentiate y=csc x (x + cot x ) . //// y’ = ....?

Question

Derivative of y

Differentiate y=csc x (x + cot x ) .

y’ = ...?

Answer 1

This problem requires use of the product rule. The product rule tells us to differentiate each factor, and sum the products of each derivative with the other factors. For the case of two products, this simplifies to the sum of the products of each factor's derivative with the other factor.

For example, the derivative of the product uv is u'v + v'u.

In this case, [(x + cot(x)) times the derivative of csc(x)] plus [csc(x) times the derivative of (x + cot(x))].

y' = (x + cot(x))(-csc(x)cot(x)) + (csc(x))(1 - csc²(x))

Now, csc²(x) = 1 + cot²(x), so -cot²(x) = 1 - csc²(x)

y' = (x + cot(x))(-csc(x)cot(x)) + (csc(x)(-cot²(x))

y' = -csc(x)cot(x)[(x + cot(x)) + cot(x)] = -csc(x)cot(x)[x + 2cot(x)]