Differentiating polynomials

Power Rule of Derivatives

• The power rule states that the derivative with respect to x of $x^n$ is equal to $n \cdot x^{n-1}$ for $n \neq 0$.

• It allows us to take the derivative of any polynomial, making it a fundamental concept in calculus.

Special Case for n = 0

• When considering the case of $n = 0$, we look at the derivative of $x^0$.

• For any $x \neq 0$, $x^0 = 1$, which means:

  • $\frac{d}{dx}(x^0) = \frac{d}{dx}(1)$

• Graphing the function $f(x) = 1$ shows it is a horizontal line on the Cartesian plane:

  • The slope of the tangent line at any point on this graph is $0$.

• Conclusion: The derivative of a constant function (such as $f(x) = a$ where $a$ is a constant) is always zero:

  • $\frac{d}{dx}(a) = 0$.

Properties of Derivatives

Scalar Multiplication Property

• If $A$ is a constant and $f(x)$ is a function, then:

  • $\frac{d}{dx}(A \cdot f(x)) = A \cdot \frac{d}{dx}(f(x))$

• Example:

  • Find the derivative of $2 \cdot x^5$:

  • Using the property: $\frac{d}{dx}(2 \cdot x^5) = 2 \cdot \frac{d}{dx}(x^5)$

  • Applying the power rule:

    • $\frac{d}{dx}(x^5) = 5x^{5-1} = 5x^4$

  • Therefore, the derivative is: $2 \cdot 5x^4 = 10x^4$.

Sum Rule of Derivatives

• For two functions $f(x)$ and $g(x)$, the derivative of their sum is:

  • $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$

• Alternative notation:

  • If $f'(x)$ denotes the derivative of $f(x)$, then:

  • $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$

• Example:

  • Find the derivative of $x^3 + x^{-4}$:

  • Using the sum rule, we differentiate each part:

    • $\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$

    • $\frac{d}{dx}(x^{-4}) = -4x^{-4-1} = -4x^{-5}$

  • Combine the results:

    • $3x^2 - 4x^{-5}$.

Composite Example Derivative Calculation

• Given $f(x) = 2x^3 - 7x^2 + 3x - 100$, we calculate:

1. $\frac{d}{dx}(2x^3)$:

   • Use scalar multiplication and power rule:

   • $2 \cdot \frac{d}{dx}(x^3) = 2 \cdot (3x^{3-1}) = 6x^2$

2. $\frac{d}{dx}(-7x^2)$:

   • $-7 \cdot \frac{d}{dx}(x^2) = -7 \cdot (2x^{2-1}) = -14x$

3. $\frac{d}{dx}(3x)$:

   • $3 \cdot \frac{d}{dx}(x) = 3 \cdot 1 = 3$

4. $\frac{d}{dx}(-100)$:

   • Derivative of any constant is $0$.

• Combining these results, we find:

  • $f'(x) = 6x^2 - 14x + 3$.

Implications and Applications

• Understanding these rules enables us to easily find the derivatives of complex polynomials and develop further into more advanced calculus topics.

• These foundational properties apply universally across a range of mathematical and engineering contexts, aiding in application problems involving rates of change, optimization processes, and graphical analysis.