​ Find d^2y/dx^2. 1. y = e^-x sin 2x 2. y = log(log x) ​

Question

Find \(\frac{d^2y}{dx^2}\). 

1. y = e^-x sin 2x 

2. y = log(log x) 

3. y = cos 4x cos 2x 

4. y = sin 3x sin 2x 

5. y = cos mx sin nx

Answer 1

1. Given y = e^-x sin 2x 

\(\frac{dy}{dx}\) = e^-x (2 cos2x) + sin 2x .(-e^-x) 

= e^-x(2 cos2x – sin2x) 

\(\frac{d^2y}{dx^2}\) = e-x (-4sin2x – 2cos2x) + (2cos2x – sin2x) (-e^-x) 

= e^-x [-4sin2x – 2cos2x – 2cos2x + sin2x] 

= e^-x (-3sin2x – 4cos2x)

2. y = log(log x)

3. y = cos 4x cpos 2x 

y = \(\frac{1}{2}\)(cos 6x + cos 2x) 

\(\frac{dy}{dx}\) = \(\frac{1}{2}\)(-6 sin6x – 2 sin2x) = -3 sin 6x – sin2x 

\(\frac{d^2y}{dx^2}\) = -18 cos 6x – 2 cos 2x 

4. Given y = sin 3x sin 2x 

y = \(\frac{1}{2}\)(cos 5x – cos 7x); 

\(\frac{dy}{dx}\) = \(\frac{1}{2}\)(-5 sin 5x + 7 sin 7x) 

\(\frac{d^2y}{dx^2}\) = \(\frac{1}{2}\)(- 25 cos 5x + 49 cos 7x) 

5. y = cosmx.sinnx 

y = \(\frac{1}{2}\)[sin (m + n)x – sin(m – n)x] 

\(\frac{dy}{dx}\) = \(\frac{1}{2}\)[(m + n)cos(m – n)x – (m – n)cos (m – n)x] 

\(\frac{d^2y}{dx^2}\) = \(\frac{1}{2}\)[-sin(m + n)x .(m + n)² + (m – n)² . sin (m – n)x] 

= \(\frac{1}{2}\)[-(m + n)² . sin (m + n)x + (m – n)² sin(m – n)x].