Derivative Rules

The Derivative of cosxsinx: A Step-by-Step Guide

Understanding the Derivative of fg

To find the derivative of fg, we must use the product rule. The product rule states that the derivative of the product of two functions f and g is given by:

(fg)' = f'g + fg'

In this case, f(x) = cosx and g(x) = sinx.

Applying the Sum and Difference Rules

The sum and difference rules of differentiation state that the derivative of the sum or difference of two functions f and g is given by:

(f ± g)' = f' ± g'

Using these rules, we can find the derivative of cosxsinx as follows:

(cosxsinx)' = (cosx)'(sinx) + (cosx)(sinx)'

= (-sinx)(sinx) + (cosx)(cosx)

= -sin^2x + cos^2x

Using the Power Rule

The power rule of differentiation states that the derivative of x^a is given by:

(x^a)' = a*x^(a-1)

Using this rule, we can find the derivative of cosx as follows:

(cosx)' = -sinx

And the derivative of sinx as follows:

(sinx)' = cosx

Combining Derivatives

Substituting the derivatives of cosx and sinx into the equation for the derivative of cosxsinx, we get:

(cosxsinx)' = (-sinx)(sinx) + (cosx)(cosx)

= -sin^2x + cos^2x

Simplifying the Result

The expression -sin^2x + cos^2x can be simplified using the trigonometric identity:

cos^2x + sin^2x = 1

Substituting this identity into the equation for the derivative of cosxsinx, we get:

(cosxsinx)' = -sin^2x + cos^2x

= 1 - 2sin^2x

Therefore, the derivative of cosxsinx is 1 - 2sin^2x.