Differenciation Help =) (1 Viewer)

[missAy]

How do you differenciate
a) 3^x

b)2^(3x-4)


Advance Pleases and Thank You(s)

 

[pLuvia]

Use the logarithmic laws
y=a^x where a is a constant
=e^ln(ax)
=e^xln(a)
dy/dx=ln(a).a^x

 

[missAy]

Thanks, now I know we haven't done that rule.

However looking at the answers shouldn't it also have the derivative of the power x as well? eg dy/dx=f'(x).ln(a).a^x ???

 

[Fish Sauce]

I'll do some long working (actually just been going over this stuff, so hopefully this is right )

a)

y = 3^x
3 = e^ln3
.:. 3^x = e^ln3.x
y' = ln3 e^ln3.x
y' = ln3.3^x

b)

y = 2^3x-4
2 = e^ln2
.:. 2^3x-4 = e^ln2(3x-4)
= e^3ln2.x - 4ln2
y' = 3ln2 e^3ln2.x - 4ln2
y' = 3ln2.2^3x-4

The derivative shouldn't be in there. What does the answer say?

 

[missAy]

I'll do some long working (actually just been going over this stuff, so hopefully this is right )

a)

y = 3^x
3 = e^ln3
.:. 3^x = e^ln3.x
y' = ln3 e^ln3.x
y' = ln3.3^x

b)

y = 2^3x-4
2 = e^ln2

I think the teacher missed a rule...I don't get WHY you do that.

.:. 2^3x-4 = e^ln2(3x-4)
= e^3ln2.x - 4ln2
y' = 3ln2 e^3ln2.x - 4ln2
y' = 3ln2.2^3x-4

The derivative shouldn't be in there. What does the answer say?

I just took it by looking at the answer that the "derivate of the power" goes out infront of the rest ie that's where I thought the 3 came from.

Yes thanks FishSauce you did get the right answer.

 

[Fish Sauce]

I think the teacher missed a rule...I don't get WHY you do that.

Basically the rule is
e^ln.x = x
So if you see e to the power of ln something then then ln cancels out the e and you're left with the number in the power.
eg. e^ln3 = 3, etc.

I just took it by looking at the answer that the "derivate of the power" goes out infront of the rest ie that's where I thought the 3 came from.

Yes thanks FishSauce you did get the right answer.

Ah okay. Haha that's good, have a test on Tuesday so I'd better know it! Hope that helped a bit.