What's the derivative of 3^x (1 Viewer)

[bettini]

How do you work it out??
f(x) = 3^x
find dy/dx

 

[Riviet]

d/dx(3^x)=d/dx(e^log_e3x)
=d/dx(e^xlog_e3)
=(e^xlog_e3).log_e3
=3^x.log_e3

 

[NickP101]

If you dont understand the above way this might help.

using y = a^x
x = log a y (a base)

so in this question

let y = 3^x

x = log 3 y (3 is the base)

change base to e as you are going to need to dy/dx later on.

x = lny / ln3
lny = x.ln3

Therefor using x = log a y (a base)

y = a^x

Hence in this question:

x.ln3 = lny

y = e^(x.ln3)

Now dy/dx of this = ln3.e^(x.ln3)

However in the beggining we let y = 3^x, and futher on we got y = e^(x.ln3), hence they are the same so the answer is:

= ln3.y
= ln3.3^x

If ive made any mistakes please correct me. Thats how i always do it, its more time consuming but always works for me.

 

[darkliight]

In general, all you need to remember is: d/dx a^x = ln(a) * a^x. (so just add the ln(a) and multiply by what you already had)

Try d/dx 4^x, -10^x and e^x if you want.

 

[Raginsheep]

You should remember the general result however, you should also be able to derive it.