How do you differentiate #f(x)=xsin(1/x)#?

Answer 1

#(d f (x))/(dx)= sin (1/x) -1/x xx cos (1/x)#

Differentiating #f (x)# is determined by applying the product rule, #" "# Chain rule ,and quotient rule. #" "# Product rule: #" "# #color (blue)((d(uv))/(dx)=(du)/(dx)xx v+ (dv)/(dx)xx u)# #" "# Chain rule says: #" "# #color (brown)((d (v (u (x))))/(dx)= (d (u(x)))/(dx) xx (dv)/(du))# #" "# Quotient rule : #" "# #color (green)(( ((u (x))/(v (x))))/(dx) = ((du (x))/(dx) xx v (x) - (dv (x))/(dx) xx u (x))/(v (x))^2# #" "# Let's start differentiating #f (x)# by applying the product rule first. #" "# #color (blue)((df (x))/(dx)= (dx)/(dx) xx sin (1/x) + (d (sin (1/x)))/(dx) xx x)# #" "# Second we should apply Chain rule on# (d (sin (1/x)))/(dx)# #" "# #(d f (x))/(dx)= (dx)/(dx) xx sin (1/x) + (color (brown)((d (1/x))/(dx) xx (d (sin (1/x)))/(d (1/x)))) xx x# #" "# Now , applying the quotient rule on#(d (1/x))/(dx)# #" "# #(d f (x))/(dx)= (dx)/(dx) xx sin (1/x) + (color (green)((0 xx x -(dx)/(dx)xx 1)/(x^2))xx (d(sin (1/x)))/(d (1/x)))) xx x# #" "# #(d f (x))/(dx)= 1 xx sin (1/x) + (-1/x^2) xx cos (1/x) xx x# #" "# #(d f (x))/(dx)= sin (1/x) -1/x xx cos (1/x)#
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Answer 2

To differentiate the function ( f(x) = x \sin\left(\frac{1}{x}\right) ), you would use the product rule. The derivative is given by:

[ f'(x) = x \cos\left(\frac{1}{x}\right) - \frac{\sin\left(\frac{1}{x}\right)}{x^2} ]

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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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