Derivative of \( \displaystyle \left(x + 2\right)^{\frac{1}{x + 2}} \)

Problem 2.494 · hard

1. \[ \frac{d}{d x} \left(x + 2\right)^{\frac{1}{x + 2}} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} e^{\frac{\ln{\left(x + 2 \right)}}{x + 2}} \]
   rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
3. \[ = e^{\frac{\ln{\left(x + 2 \right)}}{x + 2}} \frac{d}{d x} \frac{\ln{\left(x + 2 \right)}}{x + 2} \]
   chainApply the chain rule for the exponential function.✓ Proved
4. \[ = \left(\ln{\left(x + 2 \right)} \frac{d}{d x} \frac{1}{x + 2} + \frac{\frac{d}{d x} \ln{\left(x + 2 \right)}}{x + 2}\right) e^{\frac{\ln{\left(x + 2 \right)}}{x + 2}} \]
   productApply the product rule to the inner expression.✓ Proved
5. \[ = \left(- \frac{\ln{\left(x + 2 \right)}}{\left(x + 2\right)^{2}} + \frac{1}{\left(x + 2\right)^{2}}\right) e^{\frac{\ln{\left(x + 2 \right)}}{x + 2}} \]
   derivative algebraDifferentiate the individual terms. Combine the terms inside the parentheses.✓ Proved
6. \[ = \frac{\left(1 - \ln{\left(x + 2 \right)}\right) e^{\frac{\ln{\left(x + 2 \right)}}{x + 2}}}{\left(x + 2\right)^{2}} \]
   algebraFactor out the common denominator.✓ Proved
7. \[ = \frac{\left(1 - \ln{\left(x + 2 \right)}\right) \left(x + 2\right)^{\frac{1}{x + 2}}}{\left(x + 2\right)^{2}} \]
   simplifySubstitute the original expression back for the exponential term.≈ Checked numerically