Derivative sinx
The derivative of sin(x) is found by applying the derivative rules
The derivative of sin(x) is found by applying the derivative rules. The derivative of sin(x) with respect to x is denoted as d/dx (sin(x)) or simply sin'(x) and is equal to cos(x).
To understand why the derivative of sin(x) is cos(x), we need to use the definition of the derivative. The derivative measures the rate at which a function changes at any given point.
The derivative of a function f(x) is defined as the limit of the difference quotient as h approaches 0:
f'(x) = lim(h->0) [f(x+h) – f(x)] / h
Let’s apply this definition to sin(x):
f(x) = sin(x)
f(x+h) = sin(x+h)
Using the difference quotient:
f'(x) = lim(h->0) [sin(x+h) – sin(x)] / h
To evaluate this limit, we can use the trigonometric identity: sin(a+b) = sin(a)cos(b) + cos(a)sin(b).
Plugging this identity into the difference quotient:
f'(x) = lim(h->0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)] / h
As h approaches 0, we can assume that sin(h) and cos(h) approach 0 as well.
Taking the limit:
f'(x) = lim(h->0) [cos(x)sin(h)] / h
= cos(x) [lim(h->0) sin(h) / h]
The limit lim(h->0) sin(h) / h is a well-known result in calculus, and it evaluates to 1.
Therefore, the derivative of sin(x) with respect to x is cos(x). In other words, sin'(x) = cos(x).
This means that the rate at which the sine function changes (its slope or gradient) is given by the cosine function.
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