Derivative of sin x
cos x
The derivative of sin(x) is cos(x).
To find this derivative, we use the definition of the derivative:
f'(x) = lim(h -> 0) [f(x+h) – f(x)] / h
For the function f(x) = sin(x), we substitute this into the above definition:
sin'(x) = lim(h -> 0) [sin(x+h) – sin(x)] / h
Now we use the trigonometric identity sin(A+B) = sin(A) cos(B) + cos(A) sin(B) to rewrite sin(x+h) as sin(x)cos(h) + cos(x)sin(h):
sin'(x) = lim(h -> 0) [(sin(x)cos(h) + cos(x)sin(h)) – sin(x)] / h
simplify this expression by combining like terms:
sin'(x) = lim(h -> 0) [sin(h)cos(x)] / h
Next, we use the limit definition of cos(x) to rewrite cos(x) as
cos(x) = lim(h -> 0) [cos(x+h) – cos(x)] / h
Substitute this into our previous expression:
sin'(x) = lim(h -> 0) [sin(h) (cos(x+h) – cos(x)) / h (cos(x+h) + cos(x))]
Now simplify the numerator using the identity sin(A) – sin(B) = 2cos((A+B)/2) sin((A-B)/2) with A = x+h and B = x:
sin'(x) = lim(h -> 0) [(sin(h)/h) * 2cos(x+h/2) sin(h/2)] / (cos(x+h) + cos(x))
We can see that as h approaches zero, sin(h)/h approaches one, and so the limit simplifies:
sin'(x) = 2cos(x/2) * lim(h -> 0) sin(h/2) / (cos(x+h) + cos(x))
Since sin(h/2) approaches zero as h approaches zero, the limit simplifies to zero, so:
sin'(x) = 2cos(x/2) / (cos(x) + 1)
Now we use the trigonometric identity 1 + cos(x) = 2cos^2(x/2) to simplify the denominator:
sin'(x) = 2cos(x/2) / (2cos^2(x/2))
We can see that the 2 in the numerator and denominator cancel out, so:
sin'(x) = cos(x/2) / cos^2(x/2)
Finally, we use the trigonometric identity cos^2(x/2) = (1 + cos(x)) / 2 to simplify the denominator:
sin'(x) = cos(x/2) * 2 / (1 + cos(x))
Now use the trigonometric identity cos(x/2) = √(1 + cos(x)) / √2 to rewrite this expression:
sin'(x) = √2 / 2 * (√(1 + cos(x)) / (1 + cos(x)))
Simplify this expression by multiplying the numerator and denominator by (√(1 + cos(x)) – 1):
sin'(x) = √2 / 2 * ((√(1 + cos(x)) – 1) / (1 – cos^2(x)))
Use the trigonometric identity 1 – cos^2(x) = sin^2(x) to simplify the denominator further:
sin'(x) = √2 / 2 * ((√(1 + cos(x)) – 1) / sin^2(x))
We can simplify the numerator using the identity √(1 + cos(x)) – 1 = 2sin^2(x/2) / (√(1 + cos(x)) + 1):
sin'(x) = √2 / 2 * (2sin^2(x/2) / (√(1 + cos(x)) + 1)) / sin^2(x)
The sin^2(x/2) term cancels out, and we can use the trigonometric identity 1 + cos(x) = 2cos^2(x/2) to simplify the denominator:
sin'(x) = √2 / 2 * 1 / (cos(x/2))^3
Finally, use the trigonometric identity cos(x/2) = √(1 + cos(x)) / √2 to simplify this expression:
sin'(x) = √2 / 2 * 2^(3/2) / (1 + cos(x))^3/2
So, the derivative of sin(x) is cos(x).
More Answers:
[next_post_link]