What is the derivative of 3^x3x?

1 Answer
Jan 4, 2016

d/dx 3^x = 3^xln(3)ddx3x=3xln(3)

Explanation:

An easy way of doing this is by using logarithmic differentiation.

To do this, we will use the following:

  1. ln(a^x) = xln(a)ln(ax)=xln(a)
  2. The chain rule
  3. d/dxln(x) = 1/xddxln(x)=1x
  4. d/dx cx = cddxcx=c

Let y = 3^xy=3x

=> ln(y) = ln(3^x) = xln(3)ln(y)=ln(3x)=xln(3)

=> d/dxln(y) = d/dxxln(3)ddxln(y)=ddxxln(3)

=> 1/y dy/dx = ln(3)1ydydx=ln(3)

=> dy/dx = yln(3) = 3^xln(3)dydx=yln(3)=3xln(3)

:. d/dx 3^x = 3^xln(3)