# Find The Derivative of Sinx° (degree) By Using The First Principal ?

See Below.

The First Principle Of Differentiation is :-

So, Using it,

So,

The Limit

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To find the derivative of ( \sin(x^\circ) ) using the first principle, we need to recall that ( x ) should be in radians for differentiation. However, if you're specifically asked to find the derivative of ( \sin(x^\circ) ) using the first principle (which is essentially the limit definition of the derivative), you would proceed as follows:

Let ( f(x) = \sin(x^\circ) ).

Using the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute ( f(x) = \sin(x^\circ) ):

[ f'(x) = \lim_{h \to 0} \frac{\sin((x + h)^\circ) - \sin(x^\circ)}{h} ]

Now, using the angle addition formula for sine, we have:

[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) ]

Let's denote ( a = x^\circ ) and ( b = h^\circ ):

[ \sin((x + h)^\circ) = \sin(x^\circ)\cos(h^\circ) + \cos(x^\circ)\sin(h^\circ) ]

Plug this back into the expression for ( f'(x) ):

[ f'(x) = \lim_{h \to 0} \frac{\sin(x^\circ)\cos(h^\circ) + \cos(x^\circ)\sin(h^\circ) - \sin(x^\circ)}{h} ]

[ = \lim_{h \to 0} \frac{\sin(x^\circ)(\cos(h^\circ) - 1) + \cos(x^\circ)\sin(h^\circ)}{h} ]

Now, as ( h \to 0 ), ( \sin(h^\circ) ) approaches ( 0 ), and ( \cos(h^\circ) - 1 ) approaches ( 0 ). So, we have:

[ f'(x) = \lim_{h \to 0} \frac{\sin(x^\circ)(0) + \cos(x^\circ)(0)}{h} ]

[ = \lim_{h \to 0} 0 = 0 ]

Therefore, the derivative of ( \sin(x^\circ) ) using the first principle is ( 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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