Question

Find \(\frac{d^2y}{dx^2}\) if y = e^x ⋅ sin 5x? 
1. 2ex (5 cos 5x + 12 sin 5x)
2. 2ex (cos 5x - sin 5x)
3. 2e^x (5 cos 5x - 12 sin 5x)
4. 2ex (cos 5x + sin 5x)

Answer 1

Correct Answer - Option 3 : 2e^x (5 cos 5x - 12 sin 5x)

CONCEPT:

• \(\frac{{d\left( {\cos x} \right)}}{{dx}} = \; - \sin x\)
• \(\frac{{d\left( {e^x} \right)}}{{dx}} = \; e^x\)
• \(\frac{d}{{dx}}\left[ {f\left( x \right) \cdot g\left( x \right)} \right] = f\left( x \right) \cdot \;\frac{{d\left\{ {g\left( x \right)} \right\}}}{{dx}}\; + \;\;g\left( x \right) \cdot \;\frac{{d\left\{ {f\left( x \right)} \right\}}}{{dx}}\)

CALCULATION:

Given: y = ex ⋅ sin 5x

Here, we have to find \(\frac{d^2y}{dx^2}\)

So, first lets find out dy/dx.

As we know that, \(\frac{d}{{dx}}\left[ {f\left( x \right) \cdot g\left( x \right)} \right] = f\left( x \right) \cdot \;\frac{{d\left\{ {g\left( x \right)} \right\}}}{{dx}}\; + \;\;g\left( x \right) \cdot \;\frac{{d\left\{ {f\left( x \right)} \right\}}}{{dx}}\)

⇒ \(\frac{dy}{dx} = e^x \cdot sin \ 5x + 5e^{x} cos 5x \)

⇒ \(\frac{d^2y}{dx^2} = 2e^x(5 \ cos \ 5x -12 \ sin \ 5x) \)

Hence, the correct option is 3.