Derivative of x^2+2*x+1

Step-by-step solution

1. 1Sum Rule
   $$\frac{d}{dx}\left[x^2+2x+1\right]=\frac{d}{dx}\left[x^2+2x\right]+\frac{d}{dx}\left[1\right]$$

   Use the sum rule: differentiate each term on its own, then add the results.

2. 2Sum Rule
   $$\frac{d}{dx}\left[x^2+2x\right]=\frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[2x\right]$$

   Use the sum rule: differentiate each term on its own, then add the results.

3. 3Power Rule
   $$\frac{d}{dx}\left[x^2\right]=2\cdot x^{2-1}$$

   The exponent is $2$. Bring it down in front as a coefficient, then reduce the power by $1$.

4. 4Constant Multiple Rule
   $$\frac{d}{dx}\left[2x\right]=2\cdot \frac{d}{dx}\left[x\right]$$

   The factor $2$ is constant — pull it outside so only the variable part needs differentiating.

5. 5Power Rule
   $$\frac{d}{dx}\left[x\right]=1$$

   $x$ is $x^1$: by the power rule, the exponent $1$ comes down and the power drops to $x^0=1$, so the derivative is $1$.

6. =Combine
   $$\frac{d}{dx}\left[2x\right]=2\cdot 1=2$$

   Multiply: $2\cdot 1=2$.

7. =Combine
   $$\frac{d}{dx}\left[x^2+2x\right]=2x+2=2\left(x+1\right)$$

   Both terms are differentiated. Add them: $2x+2=2\left(x+1\right)$.

8. 8Constant Rule
   $$\frac{d}{dx}\left[1\right]=0$$

   This term doesn't involve $x$ at all — it's constant, so its derivative is $0$.

9. =Combine
   $$\frac{d}{dx}\left[x^2+2x+1\right]=2\left(x+1\right)+0=2\left(x+1\right)$$

   Both terms are differentiated. Add them: $2\left(x+1\right)+0=2\left(x+1\right)$.

10. ✓Final Answer
    $$\frac{d}{dx}\left[x^2+2x+1\right]=2\left(x+1\right)$$

    The first derivative simplifies to $2\left(x+1\right)$.