Find the derivative of the function.
f(x) = arctan (x/a)
Use implicit differentiation:
y=arctan(x/a)
tan(y)=x/a
sec²(y)*dy/dx = 1/a
dy/dx = (1/a)/sec²(x)
=(1/a)/(1+tan²(x))
=(a/a²)/(1+(x/a)²)
=a/(a²+x²)
To find the derivative of the function f(x) = arctan(x/a), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), the derivative of the composite function is the derivative of f(g(x)) multiplied by the derivative of g(x).
In this case, g(x) = x/a, and f(x) = arctan(g(x)). So, we need to find the derivatives of f(g(x)) and g(x) separately and then multiply them together.
Let's start with g(x):
g(x) = x/a
To find the derivative of g(x), we can use the power rule of differentiation. The power rule states that if we have a function in the form f(x) = ax^n, the derivative is f'(x) = anx^(n-1).
In this case, g(x) = x/a, which can be rearranged as g(x) = (1/a)x^1. Using the power rule, we can find the derivative of g(x):
g'(x) = (1/a)(1)x^(1-1) = 1/a
Now, let's find the derivative of f(g(x)), where f(x) = arctan(x):
f(g(x)) = arctan(g(x))
To find the derivative, we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), the derivative of the composite function is the derivative of f(g(x)) multiplied by the derivative of g(x).
In this case, f(x) = arctan(x), so the derivative of f(x) with respect to x is 1/(1 + x^2). We can substitute g(x) back into f'(x) to get the derivative of f(g(x)):
f'(g(x)) = 1/(1 + (g(x))^2)
f'(g(x)) = 1/(1 + (x/a)^2)
Now, let's multiply the derivatives together:
f'(x) = g'(x) * f'(g(x))
f'(x) = (1/a) * 1/(1 + (x/a)^2)
Simplifying, we get:
f'(x) = (1/a) / (1 + x^2/a^2)
Therefore, the derivative of f(x) = arctan(x/a) is f'(x) = (1/a) / (1 + x^2/a^2).