How do I us the Limit definition of derivative on #f(x)=sin(x)#?
Here's a video demonstrating how to use the Limit definition. You need to know the expansion
Differentiating sin(x) from first principles
By signing up, you agree to our Terms of Service and Privacy Policy
To use the limit definition of derivative on the function f(x) = sin(x), you follow these steps:

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

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

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) ]

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} ]

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} ]

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) ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 What is the equation of the line normal to # f(x)=x/ln(2x^3x)# at # x=1#?
 At what points on the graph of #f(x)=2x^39x^212x+5# is the slope of the tangent line 12?
 How do you find the equation of the tangent line to the graph of #f(x)= (ln x)^2# at x=6?
 What is the equation of the line normal to #f(x)= x^3/e^x # at #x=2#?
 How do you find the equation of the tangent and normal line to the curve #y=x^2+2x+3# at x=1?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7