Compute the derivative:

cos(x) + sin(x)

 

Possible derivation:
d/dx(cos(x) + sin(x))
Differentiate the sum term by term:
 = d/dx(cos(x)) + d/dx(sin(x))
The derivative of cos(x) is -sin(x):
 = d/dx(sin(x)) + -sin(x)
The derivative of sin(x) is cos(x):
Answer: |= -sin(x) + cos(x)

 

sin(x)√sin(x) + cos(x)√cos(x)

 

Possible derivation:
d/dx(cos^(3/2)(x) + sin^(3/2)(x))
Differentiate the sum term by term:
 = d/dx(cos^(3/2)(x)) + d/dx(sin^(3/2)(x))
Using the chain rule, d/dx(cos^(3/2)(x)) = ( du^(3/2))/( du) 0, where u = cos(x) and ( d)/( du)(u^(3/2)) = (3 sqrt(u))/2:
 = d/dx(sin^(3/2)(x)) + (3 sqrt(cos(x)) d/dx(cos(x)))/(2)
The derivative of cos(x) is -sin(x):
 = d/dx(sin^(3/2)(x)) + -sin(x) (3 sqrt(cos(x)))/2
Simplify the expression:
 = d/dx(sin^(3/2)(x)) - 3/2 sqrt(cos(x)) sin(x)
Using the chain rule, d/dx(sin^(3/2)(x)) = ( du^(3/2))/( du) 0, where u = sin(x) and ( d)/( du)(u^(3/2)) = (3 sqrt(u))/2:
 = -3/2 sqrt(cos(x)) sin(x) + (3 d/dx(sin(x)) sqrt(sin(x)))/(2)
The derivative of sin(x) is cos(x):
Answer: |= -3/2 sqrt(cos(x)) sin(x) + cos(x) (3 sqrt(sin(x)))/2

 

cos(x)√sinx + sin(x)√cos(x)

 

Possible derivation:
d/dx(cos(x) sqrt(sin(x)) + sqrt(cos(x)) sin(x))
Differentiate the sum term by term:
 = d/dx(cos(x) sqrt(sin(x))) + d/dx(sqrt(cos(x)) sin(x))
Use the product rule, d/dx(u v) = v ( du)/( dx) + u ( dv)/( dx), where u = cos(x) and v = sqrt(sin(x)):
 = d/dx(sqrt(cos(x)) sin(x)) + cos(x) d/dx(sqrt(sin(x))) + d/dx(cos(x)) sqrt(sin(x))
Simplify the expression:
 = cos(x) (d/dx(sqrt(sin(x)))) + d/dx(sqrt(cos(x)) sin(x)) + (d/dx(cos(x))) sqrt(sin(x))
Using the chain rule, d/dx(sqrt(sin(x))) = ( dsqrt(u))/( du) 0, where u = sin(x) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
 = d/dx(sqrt(cos(x)) sin(x)) + (d/dx(cos(x))) sqrt(sin(x)) + (d/dx(sin(x)))/(2 sqrt(sin(x))) cos(x)
The derivative of sin(x) is cos(x):
 = d/dx(sqrt(cos(x)) sin(x)) + (d/dx(cos(x))) sqrt(sin(x)) + cos(x) (cos(x))/(2 sqrt(sin(x)))
Simplify the expression:
 = d/dx(sqrt(cos(x)) sin(x)) + (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(cos(x))) sqrt(sin(x))
Use the product rule, d/dx(u v) = v ( du)/( dx) + u ( dv)/( dx), where u = sqrt(cos(x)) and v = sin(x):
 = (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(cos(x))) sqrt(sin(x)) + sqrt(cos(x)) d/dx(sin(x)) + d/dx(sqrt(cos(x))) sin(x)
Simplify the expression:
 = sqrt(cos(x)) (d/dx(sin(x))) + (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(cos(x))) sqrt(sin(x)) + (d/dx(sqrt(cos(x)))) sin(x)
The derivative of sin(x) is cos(x):
 = (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(cos(x))) sqrt(sin(x)) + (d/dx(sqrt(cos(x)))) sin(x) + cos(x) sqrt(cos(x))
Simplify the expression:
 = cos^(3/2)(x) + (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(cos(x))) sqrt(sin(x)) + (d/dx(sqrt(cos(x)))) sin(x)
The derivative of cos(x) is -sin(x):
 = cos^(3/2)(x) + (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(sqrt(cos(x)))) sin(x) + -sin(x) sqrt(sin(x))
Simplify the expression:
 = cos^(3/2)(x) + (cos^2(x))/(2 sqrt(sin(x))) + (d/dx(sqrt(cos(x)))) sin(x) - sin^(3/2)(x)
Using the chain rule, d/dx(sqrt(cos(x))) = ( dsqrt(u))/( du) 0, where u = cos(x) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
 = cos^(3/2)(x) + (cos^2(x))/(2 sqrt(sin(x))) - sin^(3/2)(x) + (d/dx(cos(x)))/(2 sqrt(cos(x))) sin(x)
The derivative of cos(x) is -sin(x):
 = cos^(3/2)(x) + (cos^2(x))/(2 sqrt(sin(x))) - sin^(3/2)(x) + -sin(x) (sin(x))/(2 sqrt(cos(x)))
Simplify the expression:
Answer: |= cos^(3/2)(x) + (cos^2(x))/(2 sqrt(sin(x))) - sin^(3/2)(x) - (sin^2(x))/(2 sqrt(cos(x)))

 

√sin(x) + √cos(x)

 

Possible derivation:
d/dx(sqrt(cos(x)) + sqrt(sin(x)))
Differentiate the sum term by term:
 = d/dx(sqrt(cos(x))) + d/dx(sqrt(sin(x)))
Using the chain rule, d/dx(sqrt(cos(x))) = ( dsqrt(u))/( du) 0, where u = cos(x) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
 = d/dx(sqrt(sin(x))) + (d/dx(cos(x)))/(2 sqrt(cos(x)))
The derivative of cos(x) is -sin(x):
 = d/dx(sqrt(sin(x))) + -sin(x) 1/(2 sqrt(cos(x)))
Using the chain rule, d/dx(sqrt(sin(x))) = ( dsqrt(u))/( du) 0, where u = sin(x) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
 = -(sin(x))/(2 sqrt(cos(x))) + (d/dx(sin(x)))/(2 sqrt(sin(x)))
The derivative of sin(x) is cos(x):
Answer: |= -(sin(x))/(2 sqrt(cos(x))) + cos(x) 1/(2 sqrt(sin(x)))

 

√sin(x) - √cos(x)

 

Possible derivation:
d/dx(-sqrt(cos(x)) + sqrt(sin(x)))
Differentiate the sum term by term and factor out constants:
 = -(d/dx(sqrt(cos(x)))) + d/dx(sqrt(sin(x)))
Using the chain rule, d/dx(sqrt(cos(x))) = ( dsqrt(u))/( du) 0, where u = cos(x) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
 = d/dx(sqrt(sin(x))) - (d/dx(cos(x)))/(2 sqrt(cos(x)))
The derivative of cos(x) is -sin(x):
 = d/dx(sqrt(sin(x))) - -sin(x) 1/(2 sqrt(cos(x)))
Simplify the expression:
 = d/dx(sqrt(sin(x))) + (sin(x))/(2 sqrt(cos(x)))
Using the chain rule, d/dx(sqrt(sin(x))) = ( dsqrt(u))/( du) 0, where u = sin(x) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
 = (sin(x))/(2 sqrt(cos(x))) + (d/dx(sin(x)))/(2 sqrt(sin(x)))
The derivative of sin(x) is cos(x):
Answer: |= (sin(x))/(2 sqrt(cos(x))) + cos(x) 1/(2 sqrt(sin(x)))