Differentiate and simplify the result: g(x) = e^x * ln(x^2 + 1) / (sqrt(sin x) + 2).

Step-by-step solution

1. 1

   Map the structure before differentiating anything. This is a quotient whose numerator is itself a product and whose denominator hides a chain rule. Naming the pieces keeps the bookkeeping honest:

   $g(x)=\frac{u(x)}{v(x)},\qquad u(x)=e^{x}\ln\left(x^{2}+1\right),\qquad v(x)=\sqrt{\sin x}+2$

   and the outer tool is the quotient rule

   $g'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{\left[v(x)\right]^{2}}$

2. 2

   Differentiate the numerator with the product rule and an inner chain rule. With $\left(e^{x}\right)'=e^{x}$ and $\left(\ln\left(x^{2}+1\right)\right)'=\dfrac{2x}{x^{2}+1}$ (chain rule on $\ln$ of $x^{2}+1$):

   $u'(x)=e^{x}\ln\left(x^{2}+1\right)+e^{x}\cdot\frac{2x}{x^{2}+1}=e^{x}\left(\ln\left(x^{2}+1\right)+\frac{2x}{x^{2}+1}\right)$

   Factoring out $e^{x}$ immediately keeps the later algebra shorter.

3. 3

   Differentiate the denominator. Write $\sqrt{\sin x}=\left(\sin x\right)^{1/2}$ and apply the chain rule; the constant $2$ contributes nothing:

   $v'(x)=\frac{1}{2}\left(\sin x\right)^{-1/2}\cos x=\frac{\cos x}{2\sqrt{\sin x}}$

4. 4

   Assemble the quotient rule. Substituting the four pieces:

   $g'(x)=\frac{e^{x}\left(\ln\left(x^{2}+1\right)+\dfrac{2x}{x^{2}+1}\right)\left(\sqrt{\sin x}+2\right)-e^{x}\ln\left(x^{2}+1\right)\cdot\dfrac{\cos x}{2\sqrt{\sin x}}}{\left(\sqrt{\sin x}+2\right)^{2}}$

5. 5

   Simplify by factoring the common $e^{x}$. Both terms of the numerator carry $e^{x}$, so pull it out:

   $g'(x)=\frac{e^{x}\left[\left(\ln\left(x^{2}+1\right)+\dfrac{2x}{x^{2}+1}\right)\left(\sqrt{\sin x}+2\right)-\dfrac{\ln\left(x^{2}+1\right)\cos x}{2\sqrt{\sin x}}\right]}{\left(\sqrt{\sin x}+2\right)^{2}}$

   No further cancellation is available, because the logarithm and the radical share no factors.

6. 6

   Note the domain and check numerically. The formula needs $\sin x>0$ — strictly positive, not merely non-negative, because $\sqrt{\sin x}$ sits in a denominator after differentiating — so $x\in(2k\pi,(2k+1)\pi)$. Comparing against a central difference of $g$ with $h=10^{-6}$ at $x=0.5,1,2,2.9$ gives $0.5943697$, $1.5124325$, $6.3252411$ and $27.3951392$ from both the formula and the numerical derivative, agreeing to seven significant figures.