Apply the quotient rule, which is:
dxdg(x)f(x)=g2(x)−f(x)dxdg(x)+g(x)dxdf(x)
f(x)=sin(xsin(x)) and g(x)=cos(xsin(x)).
To find dxdf(x):
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Let u=xsin(x).
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The derivative of sine is cosine:
dudsin(u)=cos(u)
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Then, apply the chain rule. Multiply by dxdxsin(x):
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Apply the product rule:
dxdf(x)g(x)=f(x)dxdg(x)+g(x)dxdf(x)
f(x)=x; to find dxdf(x):
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Apply the power rule: x goes to 1
g(x)=sin(x); to find dxdg(x):
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The derivative of sine is cosine:
dxdsin(x)=cos(x)
The result is: xcos(x)+sin(x)
The result of the chain rule is:
(xcos(x)+sin(x))cos(xsin(x))
To find dxdg(x):
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Let u=xsin(x).
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The derivative of cosine is negative sine:
dudcos(u)=−sin(u)
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Then, apply the chain rule. Multiply by dxdxsin(x):
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Apply the product rule:
dxdf(x)g(x)=f(x)dxdg(x)+g(x)dxdf(x)
f(x)=x; to find dxdf(x):
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Apply the power rule: x goes to 1
g(x)=sin(x); to find dxdg(x):
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The derivative of sine is cosine:
dxdsin(x)=cos(x)
The result is: xcos(x)+sin(x)
The result of the chain rule is:
−(xcos(x)+sin(x))sin(xsin(x))
Now plug in to the quotient rule:
cos2(xsin(x))(xcos(x)+sin(x))sin2(xsin(x))+(xcos(x)+sin(x))cos2(xsin(x))