Question
Question: What is the derivative of \[\arctan \left( {\dfrac{1}{x}} \right)\] ?...
What is the derivative of arctan(x1) ?
Solution
Hint : Here the arc function is the inverse function. Also we know the value of tan inverse function. Only the function on which the tan inverse is operated is not x as it is reciprocal of x we can say. But we will use the same formula as we use for the derivative of tan inverse.
Formula used:
dxdtan−1(x)=1+x21
Complete step by step solution:
Given the function is,
arctan(x1)
This is nothing but,
arctan(x1)=tan−1(x1)
We have to find the derivative of the function so given that is
=dxdtan−1(x1)
We know that, dxdtan−1(x)=1+x21
So here we can write x=x1 ,
=dxdtan−1(x1)
=1+(x1)21
Now taking the square,
=1+x211
Taking the LCM we can write,
=x2x2+11
Now the denominator of the fraction will shift to the numerator,
=1+x2x2
So we get the correct answer,
So, dxdtan−1(x1)=1+x2x2
So, the correct answer is “ 1+x2x2 ”.
Note : Note that the function so given is the regular function only the tan operated function is changed. Also note that all the trigonometric functions have their own inverse function and their respective derivatives also. Whenever the function is other than x ; make changes as we have done above.