For x>0 find and simplify the derivative of f(x)= arctan(x) + arctan(1/x).
I've done the problem a few times and I keep getting
(1)/(1+(x)^2)+(1)/(1+(1/(1/x)^2))(-x)^-2
but something about the answer is wrong and I cannot figure out what.
arctan(1/x) = arccot(x)
f' = 1/(1+x^2) + -1/(1+x^2) = 0
why?
because arccot(x) = pi/2 - arctan(x)
f(x) = pi/2
To find the derivative of f(x) = arctan(x) + arctan(1/x), we can use the chain rule and the formula for the derivative of arctan(x). Let's break down the process step by step.
Step 1: Derivative of arctan(x)
The derivative of arctan(x) can be found using the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), the derivative is given by f'(g(x)) * g'(x).
The derivative of arctan(x) is 1 / (1 + x^2). This can be obtained by differentiating the arctan function with respect to x.
Step 2: Derivative of arctan(1/x)
To find the derivative of arctan(1/x), we can again use the chain rule. Let's define g(x) = 1/x.
Using the chain rule, we have: d/dx arctan(g(x)) = 1 / (1 + g(x)^2) * g'(x).
The derivative of g(x) = 1/x with respect to x can be found using the quotient rule, which states that if we have a function f(x) = u(x) / v(x), the derivative is given by (u'(x)v(x) - u(x)v'(x)) / (v(x))^2.
For g(x) = 1/x, we have u(x) = 1 and v(x) = x. Therefore, g'(x) = (0 * x - 1 * 1) / x^2 = -1 / x^2.
So, d/dx arctan(1/x) = 1 / (1 + (1/x)^2) * (-1 / x^2).
Step 3: Simplify the expression
Now let's simplify the expression by combining the two derivatives:
d/dx f(x) = d/dx (arctan(x) + arctan(1/x))
= 1 / (1 + x^2) + 1 / (1 + (1/x)^2) * (-1 / x^2)
Now, to simplify the expression further, we need to get a common denominator.
Multiplying the first term by (1 + (1/x)^2) / (1 + (1/x)^2) gives: (1 / (1 + x^2)) * (1 + (1/x)^2) = (1 + (1/x)^2) / (1 + x^2).
Multiplying the second term by x^2 / x^2 gives: (1 / (1 + (1/x)^2)) * (-1 / x^2) * x^2 = -1 / (x^2 + 1/x^2).
Now, with a common denominator, we can combine the terms:
d/dx f(x) = (1 + (1/x)^2) / (1 + x^2) - 1 / (x^2 + 1/x^2)
= (1 + 1/x^2) / (1 + x^2) - 1 / (x^2 + 1/x^2).
Therefore, the simplified expression for the derivative of f(x) = arctan(x) + arctan(1/x) is:
d/dx f(x) = (1 + 1/x^2) / (1 + x^2) - 1 / (x^2 + 1/x^2).
Please double-check the steps and calculations for any mistakes that may have occurred.