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.