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

Problem 2.321 · medium

1. \[ \frac{d}{d x} \left(x + 2\right)^{x + 2} \]
   derivative rewriteStart with the derivative of the function. Rewrite the base and exponent using the exponential and logarithm functions.✓ Proved
2. \[ = \frac{d}{d x} e^{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]
   simplifySimplify the expression inside the derivative.≈ Checked numerically
3. \[ = e^{\left(x + 2\right) \ln{\left(x + 2 \right)}} \frac{d}{d x} \left(x + 2\right) \ln{\left(x + 2 \right)} \]
   chainApply the chain rule.✓ Proved
4. \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)\right) e^{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]
   productApply the product rule to the inner expression.✓ Proved
5. \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)}\right) e^{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]
   derivativeDifferentiate the first part of the product.✓ Proved
6. \[ = \left(\ln{\left(x + 2 \right)} + 1\right) e^{\left(x + 2\right) \ln{\left(x + 2 \right)}} \]
   derivative algebraDifferentiate the second part of the product. Simplify the expression inside the parentheses.✓ Proved
7. \[ = \left(x + 2\right)^{x + 2} \left(\ln{\left(x + 2 \right)} + 1\right) \]
   rewriteConvert the exponential form back to the original power form.≈ Checked numerically