d/dx (sinx)=
cosx
The derivative of sin(x) with respect to x is cos(x).
To show this, we can use the definition of the derivative:
f'(a) = lim(h → 0) [f(a + h) – f(a)]/h
In this case, f(x) = sin(x), so we have:
sin'(x) = lim(h → 0) [sin(x + h) – sin(x)]/h
Using the formula for the difference of two sine functions:
sin(a + b) – sin(a – b) = 2cos(a)sin(b)
We can rewrite the equation as:
sin'(x) = lim(h → 0) [2cos(x)(sin(h/2))/h]
As h approaches zero, sin(h/2)/h approaches 1/2. Therefore, we have:
sin'(x) = lim(h → 0) [2cos(x)(1/2)]
sin'(x) = cos(x)
So, the derivative of sin(x) with respect to x is cos(x).
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