How would you find the equation of the tangent line of f(x)=(4^x)/( x^2) at x=2

Answer 1

f(x) = 4^x / x²

Using the Quotient Rule, f'(x) = [x²(ln4)4^x - 2x(4^x)] / x⁴

Slope of tangent line when x = 2 is f'(2) = [64ln4 - 64] / 16 = 4ln4 - 4

Point on tangent line = (2, f(2)) = (2, 4)

Equation of tangent line: y - 4 = (4ln4 - 4)(x - 2)

y - 4 = 1.55(x - 2)

y = 1.55x + 0.9