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What is the derivative of #5^tanx#?

Answer 1

Emma Aldridge

#=> (dy)/(dx) = 5^tanx * ln5 * sec^2 x #

We are trying to find #(dy)/(dx) # when #y = 5^tanx #

Using both sides of natural logs...

#=> ln y = ln 5^tanx #

Applying log laws:

#alpha log beta -= log beta ^ alpha #

#=> ln y =ln5 * tanx #

Implicit differentiation:

#=> 1/y * (dy)/(dx) = ln5 * sec^2 x #

#=> (dy)/(dx) = y ln5 * sec^2 x #

#=> (dy)/(dx) = 5^tanx * ln5 * sec^2 x #

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Answer 2

William Abdalla

The derivative of (5^{\tan(x)}) with respect to (x) is (5^{\tan(x)} \cdot \ln(5) \cdot \sec^2(x) \cdot \tan(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.