Photo Math GPT

Use the chain rule to find the derivative of the function f(x) = cos(3x^2 + 2x + 1).

Prove by induction that the sum of the squares of first 'n' natural numbers is given by (n(n+1)(2n+1))/6.

Let p be a prime number. Show that for every integer a such that 0 < a < p, there exists an integer b such that b^2 ≡ a (mod p). This type of a is known as a quadratic residue modulo p.

Given the function f(x) = (3x^2 - 2x + 5)/(e^x -1), find the limit as x approaches 0.