k(x)=sin x cos x
think:
k(x)= (sin x)(cos x) , and use the product rule
What is the derivative of k(x)=sin x cos x?
A. -sin x cos x
B. -2 sin x cos x
C. 2 cos^2x-1
D. sin^2x - cos^2x
2 answers
Use the product rule,
d(sin(x)cos(x))/dx
=cos(x)dsin(x)/dx+sin(x)dcos(x)/dx
=cos²(x)-sin²(x)
Next use sin²(x)+cos²(x)=1 to transform the above result to one of the provided answers.
Note:
an easier way to derive the given expression is to make use of the identity:
sin(2x)=2sin(x)cos(x)
so
d(sin(2x))/dx
=d((1/2)sin(2x))/dx
=cos(2x)
(which equals cos²(x)-sin²(x) AND one of the posted answers)
d(sin(x)cos(x))/dx
=cos(x)dsin(x)/dx+sin(x)dcos(x)/dx
=cos²(x)-sin²(x)
Next use sin²(x)+cos²(x)=1 to transform the above result to one of the provided answers.
Note:
an easier way to derive the given expression is to make use of the identity:
sin(2x)=2sin(x)cos(x)
so
d(sin(2x))/dx
=d((1/2)sin(2x))/dx
=cos(2x)
(which equals cos²(x)-sin²(x) AND one of the posted answers)
5 answers