Let f be a function defined by f(x)= arctan x/2 + arctan x. the value of f'(0) is?
It's 3/2 but I am not very clear on how to obtain the answer. I changed arctan x/2 into dy/dx=(4-2x)/(4sqrt(4+x^2)) but that's as far as I got. Could you please show me how to solve this problem? I would really appreciate it?
Start with this rule:
d/dx (arctan x)
= 1/(1 +x^2)
d/dx (arctan x/2) = (1/2)/[1 + (x/2)^2]
Evaluate them at x = 0 and add them. You will get 3/2.
Let's differentiate both terms in the function f(x) = arctan(x/2) + arctan(x) separately and then add the results.
We'll start with arctan(x/2). The derivative of arctan(x/2) with respect to x is:
d/dx (arctan x/2) = (1/2) / [1 + (x/2)^2] = (1/2) / [1 + x^2/4].
Now, let's differentiate arctan(x) with respect to x:
d/dx (arctan x) = 1 / (1 + x^2).
To find f'(x), we need to add the derivatives of the two terms:
f'(x) = [(1/2) / (1 + x^2/4)] + [1 / (1 + x^2)].
Now, let's find f'(0) by substituting x = 0:
f'(0) = [(1/2) / (1 + 0)] + [1 / (1 + 0)] = (1/2) + 1 = 3/2.
So, f'(0) = 3/2.
To find the derivative of the function f(x) = arctan(x/2) + arctan(x), we need to use the chain rule.
First, let's find the derivative of arctan(x/2).
Using the chain rule, we have:
d/dx (arctan(x/2)) = 1 / (1 + (x/2)^2) * d/dx (x/2)
Simplifying the expression, we have:
d/dx (arctan(x/2)) = 1 / (1 + (x/2)^2) * (1/2)
Now, let's find the derivative of arctan(x).
Using the chain rule, we have:
d/dx (arctan(x)) = 1 / (1 + x^2) * d/dx (x)
Simplifying the expression, we have:
d/dx (arctan(x)) = 1 / (1 + x^2)
Now, let's find the derivative of f(x).
d/dx (f(x)) = d/dx (arctan(x/2)) + d/dx (arctan(x))
Substituting the values we found earlier, we have:
d/dx (f(x)) = (1 / (1 + (x/2)^2) * (1/2)) + (1 / (1 + x^2))
Evaluate the derivative at x = 0, we have:
d/dx (f(x))|(x=0) = (1 / (1 + (0/2)^2) * (1/2)) + (1 / (1 + 0^2))
Simplifying further, we have:
d/dx (f(x))|(x=0) = (1 / (1 + 0)) * (1/2) + (1 / 1)
d/dx (f(x))|(x=0) = (1/2) + 1
d/dx (f(x))|(x=0) = 3/2
Therefore, the value of f'(0) is 3/2.
To find the derivative of the given function f(x) = arctan(x/2) + arctan(x), you can use the formula for the derivative of the arctan function, which is:
d/dx(arctan(x)) = 1 / (1 + x^2).
First, let's find the derivative of arctan(x/2):
d/dx (arctan(x/2)) = 1 / (1 + (x/2)^2).
This simplifies to:
d/dx (arctan(x/2)) = 2 / (4 + x^2).
Now let's find the derivative of arctan(x):
d/dx(arctan(x)) = 1 / (1 + x^2).
Next, evaluate both derivatives at x = 0:
d/dx(arctan(x/2)) evaluated at x = 0 is equal to 2 / (4 + 0^2) = 2 / 4 = 1/2.
d/dx(arctan(x)) evaluated at x = 0 is equal to 1 / (1 + 0^2) = 1 / 1 = 1.
Finally, add the two derivatives:
1/2 + 1 = 3/2.
Therefore, the value of f'(0) is 3/2.