Differentiation

Question

y= (x^2) / cos4x

a) find dy/dx

b) find the approximate change in y when x is increased from pi/4 to pi/4 + p, where p is small

Answer 1

The change in y (Δy) is approximately equal to dy for values close to x.

dy/dx = f'(x) = (2xcos(4x) + 4x²sin(4x))/cos²(4x) using the quotient rule

dy = dx[(2xcos(4x) + 4x²sin(4x))/cos²(4x)]

p equals Δx ≈ dx

dy = p[(2xcos(4x) + 4x²sin(4x))/cos²(4x)]

for x = π/4, we get dy = p[(2(π/4)cos(4(π/4)) + 4(π/4)²sin(4(π/4)))/cos²(4(π/4))]

dy = p[((π/2)cos(π)) + (π²/4)sin(π)))/cos²(π)]

dy = p[-π/2 + 0)/1]

dy = -pπ/2