Math CentralQuandaries & Queries


Question from tarun, a teacher:

derivative of x^x - 2^sinx

Hi Tarun,

First differentiate the two terms separately.

When you have an expression of the form

\[y = f(x)^{g(x)}\]

where $g(x)$ is not a constant you can take the natural logarithm of both sides to obtain

\[\ln(y) = \ln\left(f(x)^{g(x)}\right) = g(x) \ln\left(f(x)\right).\]

Now differentiate both sides. The left side becomes $\large \frac{y^{\prime}}{y}$ and hence $y^{\prime}$ is $y$ times the derivative of the right side.


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