Let
f (x) = tan^(-1)((7^(x)))
f '(x) =
To find the derivative of the function f(x) = tan^(-1)(7^x), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative can be found by taking the derivative of the outer function (f') evaluated at the inner function (g(x)), multiplied by the derivative of the inner function (g'(x)).
In this case, we can let g(x) = 7^x. To find the derivative of g(x), we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = a^x, where 'a' is a constant, then the derivative of f(x) is given by f'(x) = a^x * ln(a), where ln(a) represents the natural logarithm of a.
Therefore, g'(x) = (7^x) * ln(7).
Now, let's find the derivative of f(x) using the chain rule. We have:
f'(x) = (d/dx) [tan^(-1)(g(x))]
= [d/d(g(x))] [tan^(-1)(g(x))] * [d/dx] [g(x)]
= [1 / (1 + (g(x))^2)] * g'(x)
Substituting g(x) = 7^x and g'(x) = (7^x) * ln(7), we get:
f'(x) = [1 / (1 + (7^x)^2)] * (7^x) * ln(7)
Simplifying further, we have:
f'(x) = [1 / (1 + 49^x)] * (7^x) * ln(7)
So, the derivative of f(x) = tan^(-1)(7^x) is f'(x) = [1 / (1 + 49^x)] * (7^x) * ln(7).