If #f(x)=sin x# and #g(x)=e^x#, why is #f'(x)=cos x# and #g'(x)=e^x#?

Answer 1

#f'(x)=cosx#, iff #x# is measured in radians, and #g'(x)=e^x# by definition.

If we have a function, #f(x)#, then we can define the derivative of that function, #f'(x)# as:
#lim_(h->0)# #(f(x+h)-f(x))/h#
Letting #f(x)=sinx#,
#f'(x)=lim_(h->0)# #(sin(x+h)-sin(x))/h#
#=lim_(h->0)# #(sinxcos h+cosxsinh -sinx)/h#
#=lim_(h->0)# #((sinxcos h)/h) + lim_(h->0)# #((cosxsin h)/h)# #-lim_(h->0)# #(sinx/h)#
#=sinx# #lim_(h->0)# #(cos h/h-1/h) + cosx# #lim_(h->0)# #(sin h/h)#
#lim_(h->0)# #(cos h/h-1/h)=0#
#lim_(h->0)# #(sin h/h)=1#
#thereforesinx# #lim_(h->0)# #(cos h/h-1/h) + cosx# #lim_(h->0)# #(sin h/h)#
#=sinx xx0+cosx xx1#
#=cosx#
Note: If #x# is not measured in radians, then the evaluation of the limits is different and #(sinx)'=0.0175cosx#.
Applying the same limit to #g(x)#, we eventually get:
#g'(x)= e^x# #lim_(h->0)# #((e^h-1)/h)#
#lim_(h->0)# #((a^h-1)/h)=lna#
#therefore e^x# #lim_(h->0)# #((e^h-1)/h)#
#=e^xlne#
#=e^x#
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Answer 2

The derivative of ( f(x) = \sin(x) ) is ( f'(x) = \cos(x) ) because the derivative of the sine function is the cosine function.

The derivative of ( g(x) = e^x ) is ( g'(x) = e^x ) because the exponential function ( e^x ) is its own derivative.

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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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