# What is the derivative of #3^x#?

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Begin by letting

#y=3^x# now take the ln of both sides.

#lny=ln3^xrArrlny=xln3# differentiate

#color(blue)"implicitly with respect to x"#

#rArr1/y dy/dx=ln3#

#rArrdy/dx=yln3# now y =

#3^xrArrdy/dx=3^xln3# This result can be

#color(blue)"generalised"# as follows.

#color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(a^x)=a^xlna)color(white)(a/a)|)))#

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Answer is

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Let

Taking log on both sides,

Differentiating both sides and applying chain rule,

Therefore,

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