Apply the quotient rule, which is:
dxdg(x)f(x)=g2(x)−f(x)dxdg(x)+g(x)dxdf(x)
f(x)=xlog(3x) and g(x)=9log(3).
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 1
g(x)=log(3x); to find dxdg(x):
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Let u=3x.
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The derivative of log(u) is u1.
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Then, apply the chain rule. Multiply by dxd3x:
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: x goes to 1
So, the result is: 3
The result of the chain rule is:
The result is: log(3x)+1
To find dxdg(x):
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The derivative of the constant 9log(3) is zero.
Now plug in to the quotient rule:
9log(3)log(3x)+1