Finding a derivative with $\log$

Question

$f(x)=\log{\frac{x}{1+\sqrt{5-x^2}}}$

I have to find the derivative of $f(x)$. Please tell me if my steps are correct.

$\frac{1+\sqrt{5-x^2}}{x}×\frac{1}{\ln 10}×\frac{1+\sqrt{5-x^2}-x×\frac{1}{2\sqrt{5-x^2}}}{(1+\sqrt{5-x^2})^2}×\frac{1}{2\sqrt{5-x^2}}×(-2x)$

Usually $\log x$ refers to the natural logarithm in real analysis. Are you sure it means $\log_{10} x$ here? — PhysicsMathsLove, Apr 22 '18 at 12:03

score 1 · Answer 1 · answered Apr 22 '18 at 12:05

1

I would write $$f(x)=\log(x)-\log(1+\sqrt{5-x^2})$$ then we get $$f'(x)=\frac{1}{x}-\frac{1}{1+\sqrt{5-x^2}}\cdot \frac{1}{2}(5-x^2)^{-1/2}\cdot (-2x)$$ if you mean $$\log(x)=\ln(x)$$ the logarithm to the base $e$

answered Apr 22 '18 at 12:05

Dr. Sonnhard Graubner

95,283

In my case $\log=\log_{10}$ – user554506 Apr 22 '18 at 12:12
Then it is $$\log_{10}{x}=\frac{\ln(x)}{\ln(10)}$$ easy to differentiate. – Dr. Sonnhard Graubner Apr 22 '18 at 12:14
I'm stuck at this point: $\frac{(1+\sqrt{5-x^2})(\sqrt{5-x^2})+x^2}{x(\ln 10)(1+\sqrt{5-x^2})(\sqrt{5-x^2})}$ How can I simplify this? – user554506 Apr 22 '18 at 12:19

Finding a derivative with $\log$

1 Answers1

Linked