(Original post by sulexk)
You have ln(3x+5), differentiating this, will give [1/(3x+5)] *3, the *3 comes from differentiating what is inside the brackets. I can explain this using the "CHAIN RULE":
Don't Worry, if you have never seen this before, I will try and explain it as simply as I can:
Let us use the above for our example:
say "y=ln(3x+5)", and I said to you "I want to differentiate this", you will say to me "you have a function (3x+5) within another function ln(). so we have a function within a function!
Suppose, I said to you let us say that u=3x+5.
then simply put y=ln(u)
now if I want to differentiate, I can only find dy/du, since the right hand side is in terms of u, but I want to find dy/dx, so what do I do?
Well now let us use some intriguing manipulation:
(dy/du) * (du/dx) = dy/dx
now how do we find dy/du and du/dx
Well, dy/du = derivative of y=ln(u) which is 1/u
we know u=3x+5, so du/dx = 3
then substituting these into the above: (dy/du) * (du/dx) = dy/dx
(1/u)*(3)=3/u as u=3x+5
then dy/dx = 3/(3x+5)
and there you have it a manipulation of the chain rule to find the derivative of y=3x+5
If you want some practice with this, feel free to try some examples below:
Find the derivative of each of the following:
1) y= 1/(4x+5)
2) y= sin(3x+5)
3) y=1/(x^2 + 1)
5)y=cos(x^2 + 1)
When you finished feel free to let me know, and I'll answer them and put up the solutions!
Practice is the way forward! If you want me to provide you with some applied problems, let me know! (I can give you some rate of change problems!).
hope that helps!!