Question

What is the derivative of log x with respect to tan-1x?
1. ​\(\rm \frac{(1+x^2)}{x^2} \)​
2. \(\rm \frac{x^2}{​​​​1+x^2} \)
3. \(\rm \frac{(1+x^2)}{x} \)
4. \(\rm \frac{x}{​​​​1+x^2} \)

Answer 1

Correct Answer - Option 3 : \(\rm \frac{(1+x^2)}{x} \)

Concept: 

Steps for derivatives of functions expressed in the parametric form:

1. First of all, we write the given functions u and v in terms of the parameter x.
2. Using differentiation find out du/dx and dv/dx.
3. Then by using the formula used for solving functions in parametric form i.e.
4.  Lastly substituting the values of du/dx and dv/dx and simplify to obtain the result.

Calculation:

Let u = log x and v = tan^-1x

Differentiating with respect to x, we get

\(\rm ⇒ \frac{du}{dx} = \frac{1}{x} \;and\; \frac{dv}{dx} = \frac{1}{1 +x^2}\)

Now,

\(\rm ⇒ \frac{d\log x}{d\tan^{-1} x} = \frac{du}{dv} = \frac{ \frac{du}{dx}}{ \frac{dv}{dx}}\)

= \(\rm \frac{(1+x^2)}{x} \)