What is the derivative of #3^x#?

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Jim G. Share
Sep 14, 2016

Answer:

#3^xln3#

Explanation:

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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Dec 11, 2017

Answer:

#d/dx(3^x)=ln(3)3^x#

Explanation:

#d/dx(3^x)#
#=d/dx(e^(x ln(3)))# (since #a^b=e^(b ln(a))#)
#=d/dx(x ln(3))d/d(x ln(3))(e^(x ln(3)))# (the chain rule)
#=ln(3)e^(x ln(3))#
#=ln(3)3^x#

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