derivative of sin(x)
cos(x)
The derivative of sin(x) is cos(x).
We can prove this by using the definition of the derivative:
f'(x) = lim Δx→0 [f(x+Δx) – f(x)]/Δx
Let f(x) = sin(x). Then, using the sum-to-product formula for sin, we have:
f(x+Δx) = sin(x)cos(Δx) + cos(x)sin(Δx)
[f(x+Δx) – f(x)]/Δx = [sin(x)cos(Δx) + cos(x)sin(Δx) – sin(x)]/Δx
= [cos(x)sin(Δx)]/Δx
As Δx approaches 0, sin(Δx)/Δx approaches 1, so we are left with:
lim Δx→0 [f(x+Δx) – f(x)]/Δx = cos(x)
Therefore, the derivative of sin(x) is cos(x).
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