Apply the quotient rule, which is:
dxdg(x)f(x)=g2(x)−f(x)dxdg(x)+g(x)dxdf(x)
f(x)=xlog(x) and g(x)=2.
To find dxdf(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 2x1
g(x)=log(x); to find dxdg(x):
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The derivative of log(x) is x1.
The result is: 2xlog(x)+x1
To find dxdg(x):
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The derivative of the constant 2 is zero.
Now plug in to the quotient rule:
4xlog(x)+2x1
Apply the quotient rule, which is:
dxdg(x)f(x)=g2(x)−f(x)dxdg(x)+g(x)dxdf(x)
f(x)=x and g(x)=x.
To find dxdf(x):
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Apply the power rule: x goes to 2x1
To find dxdg(x):
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Apply the power rule: x goes to 1
Now plug in to the quotient rule:
−2x231