How do I us the Limit definition of derivative on #f(x)=sin(x)#?

Answer 1

Here's a video demonstrating how to use the Limit definition. You need to know the expansion #sin(a+b)=sinacosb+cosasinb# to complete the proof.

Differentiating sin(x) from first principles

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Answer 2

To use the limit definition of derivative on the function f(x) = sin(x), you follow these steps:

  1. Start with the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

  2. Substitute the function f(x) = sin(x) into the definition: [ f'(x) = \lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h} ]

  3. Apply the trigonometric identity for the difference of angles: [ \sin(A + B) - \sin(A) = 2 \cos \left( A + \frac{B}{2} \right) \sin \left( \frac{B}{2} \right) ]

  4. Use this identity to simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{2 \cos \left( x + \frac{h}{2} \right) \sin \left( \frac{h}{2} \right)}{h} ]

  5. Cancel out the common factor of ( \sin \left( \frac{h}{2} \right) ) from numerator and denominator: [ f'(x) = \lim_{h \to 0} \frac{2 \cos \left( x + \frac{h}{2} \right)}{1} ]

  6. Evaluate the limit as ( h ) approaches 0: [ f'(x) = 2 \cos(x) ]

Therefore, the derivative of ( f(x) = \sin(x) ) is ( f'(x) = 2 \cos(x) ).

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Answer from HIX Tutor

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.

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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