Question
Find the second order derivatives of the following functions:$\log(\sin\text{x})$
✓
Answer
We have,
$\text{y}=\text{e}^\text{e}\sin(5\text{x})$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}\sin5\text{x}+\text{e}^\text{x}\cos5\text{x}\times5$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\text{e}^\text{x}\sin5\text{x}+\text{e}^\text{x}\cos5\text{x}\times5+5\text{e}^\text{x}\cos5\text{x}$
$=-24\text{e}^\text{x}\sin5\text{x}+10\text{e}^\text{x}\cos5\text{x}$
$=2\text{e}^\text{x}(5\cos5\text{x}-12\sin5\text{x})$
$\text{y}=\text{e}^\text{e}\sin(5\text{x})$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}\sin5\text{x}+\text{e}^\text{x}\cos5\text{x}\times5$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\text{e}^\text{x}\sin5\text{x}+\text{e}^\text{x}\cos5\text{x}\times5+5\text{e}^\text{x}\cos5\text{x}$
$=-24\text{e}^\text{x}\sin5\text{x}+10\text{e}^\text{x}\cos5\text{x}$
$=2\text{e}^\text{x}(5\cos5\text{x}-12\sin5\text{x})$
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