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How do you find the derivative of #x^tanx#?

Answer 1

#x^{tan(x)}(ln(x)*sec^{2}(x)+tan(x)/x)#

Use logarithmic differentiation: let #y=x^{tan(x)}# so that #ln(y)=ln(x^{tan(x)})=tan(x)ln(x)#.

Now differentiate both sides with respect to #x#, keeping in mind that #y# is a function of #x# and using the Chain Rule and Product Rule:

#1/y * dy/dx=sec^{2}(x)ln(x)+tan(x)/x#

Hence,

#dy/dx=y * (ln(x)sec^{2}(x)+tan(x)/x)#

#=x^{tan(x)} (ln(x)sec^{2}(x)+tan(x)/x)#

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Answer 2

Cameron Alexander

To find the derivative of ( x^{\tan(x)} ), you can use logarithmic differentiation.

[ \frac{d}{dx} \left(x^{\tan(x)}\right) = x^{\tan(x)} \left(\tan(x) \cdot \frac{1}{x} + \ln(x) \cdot \sec^2(x) \cdot \ln(x)\right) ]

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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.