Prove that the function f(x) = 10x^7 + 7^x is increasing.

In order to verify if the function f(x) is increasing,
we'll have to demonstrate that the first derivative of the function is
positive.

Let's calculate f'(x) = (10x^7 +
7^x)'

We'll re-write
f'(x):

f'(x) = (10x^7)' +
(7^x)'

We'll re-write
f'(x):

f'(x) = 70x^6 +
7^x*ln7

Since 70x^6 >0 and 7^x*ln7 > 0
(7^x>0 and ln 7 = 1.94) , then 70x^6 + 7^x*ln7 >
0

The expression of f'(x) is positive, for
any value of x, so f(x) is an increasing
function.