What is the derivative of y=(sinx)^x?
3 Answers
Explanation:
Use logarithmic differentiation.
Differentiate implicitely: (Use the product rule and the chain ruel)
So, we have:
Solve for
Explanation:
The easiest way to see this is using:
(sinx)^x=e^(ln((sinx)^x))=e^(xln(sinx))
Taking the derivative of this gives:
d/dx(sinx)^x=(d/dxxln(sinx))e^(xln(sinx))
=(ln(sinx)+xd/dx(ln(sinx)))(sinx)^x
=(ln(sinx)+x(d/dxsinx)/sinx)(sinx)^x
=(ln(sinx)+xcosx/sinx)(sinx)^x
=(ln(sinx)+xcotx)(sinx)^x
Now we must note that if
However, when we analyse the behaviour of the function around the
(sinx)^x approaches 0
then:
ln((sinx)^x) will approach-oo
so:
e^(ln((sinx)^x)) will approach 0 as well
Furthermore, we note that if
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