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