Calculate the value of the derivative of the function f (x) at a given point: f (x) = 5 ^ x / x ^ 2 + 1, f ‘(1)

Let us calculate the derivative of the function at the point f ‘(1), where f (x) = 5 ^ x / (x ^ 2 + 1).

1) First, we find the derivative of the function.

In order to find the derivative of a function, we use the derivative formulas:

(x – y) ‘= x’ – y ‘;
(x ^ n) ‘= n * x ^ (n – 1);
(a ^ x) ‘= a ^ x * ln a;
x ‘= 1;
C ‘= 0.
Then we get:

f ‘(x) = (5 ^ x / (x ^ 2 + 1))’ = (5 ^ x * ln 5 * (x ^ 2 + 1) – 5 ^ x * (2 * x + 0)) / (x ^ 2 + 1) ^ 2 = (5 ^ x * ln 5 * (x ^ 2 + 1) – 2 * x * 5 ^ x) / (x ^ 2 + 1);

2) Find the value of the derivative of the function at the point x = 1.

f (1) = (5 ^ 1 * ln 5 * (1 ^ 2 + 1) – 2 * 1 * 5 ^ 1) / (1 ^ 2 + 1) = (5 * 1.6 * 2 – 2 * 5) / 2 = 5 * 1.6 – 5 = 5 * (1.6 – 1) = 5 * 0.6 = 3;

So f ‘(1) = 3.