Apply the quotient rule, which is:
dxdg(x)f(x)=g2(x)−f(x)dxdg(x)+g(x)dxdf(x)
f(x)=tan(x) and g(x)=x.
To find dxdf(x):
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Rewrite the function to be differentiated:
tan(x)=cos(x)sin(x)
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Apply the quotient rule, which is:
dxdg(x)f(x)=g2(x)−f(x)dxdg(x)+g(x)dxdf(x)
f(x)=sin(x) and g(x)=cos(x).
To find dxdf(x):
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The derivative of sine is cosine:
dxdsin(x)=cos(x)
To find dxdg(x):
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The derivative of cosine is negative sine:
dxdcos(x)=−sin(x)
Now plug in to the quotient rule:
cos2(x)sin2(x)+cos2(x)
To find dxdg(x):
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Apply the power rule: x goes to 1
Now plug in to the quotient rule:
x2cos2(x)x(sin2(x)+cos2(x))−tan(x)