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What is the derivative of #x^(cosx)#?

Answer 1

Ezekiel Alexander

We shall attack this problem by logarithmic differentiation.

Let, #y = x^(cos x)#

Taking natural logarithm on both the sides, we obtain

# Ln y = Cos x Ln x#

Now, differentiating with respect to #x#,

#1/y(dy)/dx = Cos x/x - Sin xLn x#

In the last step, I used chain rule on the LHS and the product rule on the RHS.

Thus, #(dy)/dx = x^Cos x [Cos x/x - Sin xLn x]#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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