Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDifferentiation3 Marks
Question
Differentiate the following w.r.t. x :$y=\frac{x e^x}{x+e^x}$
✓
Answer
$y=\frac{x e ^x}{x+ e ^x}$
Differentiating w.r.t. $x$, we get
$\frac{ d y}{ d x}=\frac{ d }{ d x}\left(\frac{x e ^x}{x+ e ^x}\right)$
$=\frac{\left(x+ e ^x\right) \frac{ d }{ d x} x e ^x-x e ^x \frac{ d }{ d x}\left(x+ e ^x\right)}{\left(x+ e ^x\right)^2}$
$=\frac{\left(x+ e ^x\right)\left(x \frac{ d }{ d x} e ^x+ e ^x \frac{ d }{ d x} x\right)-x e ^x\left(\frac{ d }{ d x} x+\frac{ d }{ d x} e ^x\right)}{\left(x+ e ^x\right)^2}$
$=\frac{\left(x+ e ^x\right)\left(x e ^x+ e ^x\right)-x e ^x\left(1+ e ^x\right)}{\left(x+ e ^x\right)^2}$
$=\frac{\left(x+ e ^x\right) e ^x(x+1)-x e ^x\left(1+ e ^x\right)}{\left(x+ e ^x\right)^2}$
$=\frac{ e ^x\left[\left(x+ e ^x\right)(x+1)-x\left(1+ e ^x\right)\right]}{\left(x+ e ^x\right)^2}$
$=\frac{ e ^x\left[x^2+x+x e ^x+ e ^x-x-x e ^x\right]}{\left(x+ e ^x\right)^2}$
$=\frac{ e ^x\left(x^2+ e ^x\right)}{\left(x+ e ^x\right)^2}$
$$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.