derivative of sinx
The derivative of sin(x) is cos(x)
The derivative of sin(x) is cos(x).
To understand why, let’s use the definition of the derivative. The derivative of a function f(x) measures the rate at which the function changes at different points. It is calculated as the limit of the difference quotient as the interval of change approaches zero.
For sin(x), we can start by writing the difference quotient:
f'(x) = lim(h->0) [(sin(x+h) – sin(x))/h]
To simplify this expression, we can use the trigonometric identity: sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
Applying this identity, we can rewrite the difference quotient:
f'(x) = lim(h->0) [(sin(x)cos(h)+cos(x)sin(h) – sin(x))/h]
Next, we can rearrange terms and factor out sin(x):
f'(x) = lim(h->0) [sin(x)(cos(h)-1)/h + cos(x)sin(h)/h]
Since sin(x) is not dependent on h, we can separate the two terms:
f'(x) = sin(x)lim(h->0) [(cos(h)-1)/h] + cos(x)lim(h->0) [sin(h)/h]
Now, as h approaches 0, both limits exist. The first limit [(cos(h)-1)/h] evaluates to 0, and the second limit [sin(h)/h] evaluates to 1.
Therefore, we can simplify the expression further:
f'(x) = sin(x)(0) + cos(x)(1)
f'(x) = cos(x)
Hence, the derivative of sin(x) is cos(x).
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