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What is the derivative of #5^(x/6)#?

Answer 1

Charles Anctil

#=((log 5)/6)5^(x/6)#

Use #a^(kx)=e^(x k log a)# and #(e^(bx))'=be^(bx)#.

#(5^(x/6))'#

#=(e^(((log 5)/6)x))'#

#=((log 5)/6)(e^(((log 5)/6)x))#

#=((log 5)/6)5^(x/6)#

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Answer 2

Emily Ammerman

The derivative of (5^{x/6}) with respect to (x) can be found using the general formula for the derivative of an exponential function (a^{u(x)}), where (a) is a constant and (u(x)) is a function of (x). The formula is:

[ \frac{d}{dx}a^{u(x)} = a^{u(x)} \ln(a) \cdot u'(x) ]

For (5^{x/6}), (a = 5) and (u(x) = \frac{x}{6}), so (u'(x) = \frac{1}{6}). Applying the formula:

[ \frac{d}{dx}5^{x/6} = 5^{x/6} \ln(5) \cdot \frac{1}{6} ]

Therefore, the derivative of (5^{x/6}) is (\frac{5^{x/6} \ln(5)}{6}).

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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.