d/dx arctan(x)
To find the derivative of the function f(x) = arctan(x), we can use the definition of the derivative:
The derivative of f(x) with respect to x, denoted as f'(x) or df/dx, is defined as the limit of the difference quotient as h approaches 0:
f'(x) = lim(h→0) (f(x+h) – f(x)) / h
Let’s compute the derivative step by step
To find the derivative of the function f(x) = arctan(x), we can use the definition of the derivative:
The derivative of f(x) with respect to x, denoted as f'(x) or df/dx, is defined as the limit of the difference quotient as h approaches 0:
f'(x) = lim(h→0) (f(x+h) – f(x)) / h
Let’s compute the derivative step by step.
1. Substitute f(x) = arctan(x) into the difference quotient:
f'(x) = lim(h→0) (arctan(x+h) – arctan(x)) / h
2. Apply the identity for the difference of arctan functions:
arctan(a) – arctan(b) = arctan((a – b) / (1 + ab))
In this case, let a = x + h and b = x:
f'(x) = lim(h→0) arctan((x + h – x) / (1 + x(x + h)))
Simplifying the expression:
f'(x) = lim(h→0) arctan(h / (1 + x(x + h)))
3. Apply a trigonometric identity to simplify the expression further:
arctan(u/v) = arctan(u) – arctan(v)
In this case, let u = h and v = (1 + x(x + h)):
f'(x) = lim(h→0) (arctan(h) – arctan(1 + x(x + h)))
4. Now, let’s focus on the two terms separately and find their derivatives:
First, the derivative of arctan(h) with respect to h is 1 / (1 + h^2).
Second, the derivative of arctan(1 + x(x + h)) with respect to h can be found using the chain rule. Let’s define g(h) = 1 + x(x + h). Then:
dg/dh = x + 2xh
So, the derivative of arctan(1 + x(x + h)) with respect to h is (1 / (1 + (1 + x(x + h))^2)) * (x + 2xh).
5. Let’s substitute the derivatives back into the difference quotient:
f'(x) = lim(h→0) ((1 / (1 + h^2)) – (1 / (1 + (1 + x(x + h))^2)) * (x + 2xh))
6. Simplify the expression further:
f'(x) = (1 / (1 + x^2)) – (1 / (1 + x^2)) * lim(h→0) (1 / (1 + (1 + x(x + h))^2)) * (x + 2xh)
7. The term lim(h→0) (1 / (1 + (1 + x(x + h))^2)) can be simplified further as h approaches 0:
lim(h→0) (1 / (1 + (1 + x(x + h))^2)) = 1 / (1 + (1 + x^2)^2)
8. Substitute this value back into the expression:
f'(x) = (1 / (1 + x^2)) – (1 / (1 + x^2)) * (x + 2x*0) / (1 + (1 + x^2)^2)
f'(x) = (1 / (1 + x^2)) – (1 / (1 + x^2)) * x / (1 + (1 + x^2)^2)
9. Simplify the expression:
f'(x) = (1 / (1 + x^2)) – (x / (1 + x^2)) / (1 + (1 + x^2)^2)
f'(x) = (1 – x) / (1 + x^2 + 2x^2 + x^4)
f'(x) = 1 – x / (1 + 3x^2 + x^4)
So, the derivative of arctan(x) with respect to x is (1 – x) / (1 + 3x^2 + x^4).
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