derivative of sinx
cosx
The derivative of sin(x) is cos(x), which means that if we have a function f(x) = sin(x), the derivative of f(x) with respect to x (i.e. df/dx) is f'(x) = cos(x).
To understand why this is the case, we can use the definition of the derivative, which tells us that:
f'(x) = lim(h -> 0) [f(x + h) – f(x)] / h
To find the derivative of sin(x), we substitute f(x) = sin(x) into this formula and simplify:
f'(x) = lim(h -> 0) [sin(x + h) – sin(x)] / h
We can apply the angle sum formula for sine to simplify the expression in the numerator:
f'(x) = lim(h -> 0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)] / h
We can then factor out sin(x) from the terms in the numerator and cancel it with the denominator:
f'(x) = lim(h -> 0) [sin(x)(cos(h) – 1) + cos(x)sin(h)] / h
As h approaches zero, the first term in the numerator simplifies to zero since cos(h) – 1 approaches zero. We can then use the limit definition of the derivative for the second term in the numerator to get:
f'(x) = cos(x)
Therefore, the derivative of sin(x) is cos(x).
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