derivative of log(chain rule)

[rbcobra]

I need to take the derivative of y=log[x+(1+x^2)^(1/2)] and show that it is dy/dx=(1+x^2)^(-1/2)



WORK DONE :

I have used the chain rule to come up with:

dy/dx=1/(x+(1+x^2))* (1+x(1+x^2)^(-1/2))

but i cannot seem to find out how it cancels out to equal just (1+x^2)^(-1/2)

any help would be greatly appreciated. thanks.

RC

 

[galactus]

The algebra is the booger, huh?. Remember back to rationalizing denominators and numerators?.

Upon appyling the chain rule, we get:

\(\displaystyle \underbrace{\left(\frac{1}{x+\sqrt{1+x^{2}}}\right)}_{\text{diff. of log}}\overbrace{\left(1+\frac{x}{\sqrt{1+x^{2}}}\right)}^{\text{diff. of inside}}\)

\(\displaystyle =\frac{x+\sqrt{1+x^{2}}}{\sqrt{1+x^{2}}(x+\sqrt{1+x^{2}})}\)

Now, rationalize the numerator by multiplying top and bottom by \(\displaystyle x-\sqrt{1+x^{2}}\)

It will simplify down to the required result.

 

[rbcobra]

oh my goodness thank you!

 

[Pklarreich]

The algebra is the booger, huh?. Remember back to rationalizing denominators and numerators?.

Upon appyling the chain rule, we get:

\(\displaystyle \underbrace{\left(\frac{1}{x+\sqrt{1+x^{2}}}\right)}_{\text{diff. of log}}\overbrace{\left(1+\frac{x}{\sqrt{1+x^{2}}}\right)}^{\text{diff. of inside}}\)

\(\displaystyle =\frac{x+\sqrt{1+x^{2}}}{\sqrt{1+x^{2}}(x+\sqrt{1+x^{2}})}\)

Now, rationalize the numerator by multiplying top and bottom by \(\displaystyle x-\sqrt{1+x^{2}}\)

It will simplify down to the required result.

Don't they just cancel?

 

[mmm4444bot]

Don't they just cancel?

Hello Pklarreich:

Your question above contains an unreferenced pronoun.

Does what cancel? To what are you trying to refer when you use the word "they"?

~ Mark :?

 

[Deleted member 4993]

The algebra is the booger, huh?. Remember back to rationalizing denominators and numerators?.

Upon appyling the chain rule, we get:

\(\displaystyle \underbrace{\left(\frac{1}{x+\sqrt{1+x^{2}}}\right)}_{\text{diff. of log}}\overbrace{\left(1+\frac{x}{\sqrt{1+x^{2}}}\right)}^{\text{diff. of inside}}\)

\(\displaystyle =\frac{x+\sqrt{1+x^{2}}}{\sqrt{1+x^{2}}(x+\sqrt{1+x^{2}})}\)

Now, rationalize the numerator by multiplying top and bottom by \(\displaystyle x-\sqrt{1+x^{2}}\)

It will simplify down to the required result.

Don't they just cancel?

Correct - no need to rationalize.