Derivative of 2*x^2+5*x-3

Step-by-step solution

1. 1Difference Rule
   $$\frac{d}{dx}\left[2\cdot x^2+5x-3\right]=\frac{d}{dx}\left[2\cdot x^2+5x\right]-\frac{d}{dx}\left[3\right]$$

   Use the difference rule: differentiate each term on its own, then subtract.

2. 2Sum Rule
   $$\frac{d}{dx}\left[2\cdot x^2+5x\right]=\frac{d}{dx}\left[2\cdot x^2\right]+\frac{d}{dx}\left[5x\right]$$

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

3. 3Constant Multiple Rule
   $$\frac{d}{dx}\left[2\cdot x^2\right]=2\cdot \frac{d}{dx}\left[x^2\right]$$

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

4. 4Power 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 $1$.

5. =Combine
   $$\frac{d}{dx}\left[2\cdot x^2\right]=2\cdot 2x=4x$$

   Multiply: $2\cdot 2x=4x$.

6. 6Constant Multiple Rule
   $$\frac{d}{dx}\left[5x\right]=5\cdot \frac{d}{dx}\left[x\right]$$

   The factor $5$ is constant — pull it outside so only the variable part needs differentiating.

7. 7Power 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 $1$.

8. =Combine
   $$\frac{d}{dx}\left[5x\right]=5\cdot 1=5$$

   Multiply: $5\cdot 1=5$.

9. =Combine
   $$\frac{d}{dx}\left[2\cdot x^2+5x\right]=4x+5=4x+5$$

   Both terms are differentiated. Add them: $4x+5=4x+5$.

10. 10Constant Rule
    $$\frac{d}{dx}\left[3\right]=0$$

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

11. =Combine
    $$\frac{d}{dx}\left[2\cdot x^2+5x-3\right]=4x+5-0=4x+5$$

    Both terms are differentiated. Subtract: $4x+5-0=4x+5$.

12. ✓Final Answer
    $$\frac{d}{dx}\left[2\cdot x^2+5x-3\right]=4x+5$$

    The first derivative simplifies to $4x+5$.