Question

find d²y/dx² if y=e*6x cos3x​

Answer 1

Step-by-step explanation:

We have,

$y = {e}^{6x} . \cos(3x)$

Differentiating both sides w.r.t x, we get,

$\frac{dy}{dx} = 6 {e}^{6x} . \cos(3x) - 3 {e}^{6x} \sin(3x)$

$\implies\frac{dy}{dx} = 3 {e}^{6x} \{2 \cos(3x) - \sin(3x) \}$

Again, differentiating both sides w.r.t x, we get,

$\implies\frac{d^{2} y}{dx^{2} } = 3.6. {e}^{6x} \{2 \cos(3x) - \sin(3x) \} +3{e}^{6x} \{ - 6 \sin(3x) - 3\cos(3x) \}$

$\implies\frac{d^{2} y}{dx^{2} } =18 {e}^{6x} \{2 \cos(3x) - \sin(3x) \} - 9{e}^{6x} \{ 2 \sin(3x) + \cos(3x) \}$

$\implies\frac{d^{2} y}{dx^{2} } =36{e}^{6x} \cos(3x) - 18 {e}^{6x} \sin(3x) - 18{e}^{6x} \sin(3x) - 9 {e}^{6x} \cos(3x)$

$\implies\frac{d^{2} y}{dx^{2} } =(36{e}^{6x} - 9 {e}^{6x} )\cos(3x) - 36 {e}^{6x} \sin(3x)$

$\implies\frac{d^{2} y}{dx^{2} } =27{e}^{6x} \cos(3x) - 36 {e}^{6x} \sin(3x)$

$\implies\frac{d^{2} y}{dx^{2} } =9{e}^{6x} \{3 \cos(3x) - 4 \sin(3x) \}$