Question

find d^2 y /dx^2 if y = e^5x - 3x^7+ 4 cos 3x​

Answer 1

${ \bold{ \underline{Given} : - }} \\ \\ \: \: y = {e}^{5x} - 3 {x}^{7} + 4 \cos(3x) \\ \\ { \bold{ \underline{ To \: \: find} : - }} \\ \\ = > \frac{ {d}^{2}y }{d {x}^{2} } \\ \\ { \bold{ \underline{solution} : - }} \\ \\ = > y = {e}^{5x} - 3 {x}^{7} + 4 \cos(3x) \\ \\ \\ \: \: \: . \: \: Now \: \: differentiate \: \: with \: \: respect \: \: to \: \: x - \\ \\ = > \frac{dy}{dx} = 5 {e}^{5x} - 21 {x}^{6} + 4 ( - \sin(3x) ) \times 3 \\ \\ \\ \: \: \: . \: \: Again \: \: differentiate \: \: with \: \: respect \: \: to \: \: x - \\ \\ = > \frac{ {d}^{2}y }{d {x}^{2} } = 25 {e}^{5x} - 126 {x}^{5} - 12 \times 3( \cos(3x) ) \\ \\ = > \frac{ {d}^{2}y }{d {x}^{2} } = 25 {e}^{5x} - 126 {x}^{5} - 36 \cos(3x) \\ \\ \\ { \bold{ \underline{Used \: \: formula} : - }} \\ \\ (1) \: \: \frac{d( {e}^{x}) }{dx} = {e}^{x} \\ \\ (2) \: \: \frac{d( {x}^{n}) }{dx} = n {x}^{n - 1} \\ \\ (3) \: \: \frac{d( \cos(x) )}{dx} = - \sin(x) \\ \\ (4) \: \: \frac{d( \sin(x)) }{dx} = \cos(x)$