# Errors and Approximations

### Errors in Using Derivative Rules:

1. Algebraic Mistakes:

• Errors can occur due to mistakes in applying derivative rules like the power rule, product rule, quotient rule, or chain rule.
• Example: Incorrectly differentiating a function such as $f\left(x\right)={x}^{4}+3{x}^{2}-2$ leading to ${f}^{\mathrm{\prime }}\left(x\right)=4{x}^{3}+6x$ instead of the correct derivative ${f}^{\mathrm{\prime }}\left(x\right)=4{x}^{3}+6x$.
2. Chain Rule Errors:

• Mistakes in applying the chain rule, especially in complex composite functions, may result in errors in derivative calculations.
• Example: Incorrectly applying the chain rule while differentiating a function like $f\left(x\right)=\mathrm{sin}\left(3{x}^{2}\right)$, leading to an incorrect derivative.

### Approximations Using Derivative Rules:

1. Linear Approximation:

• Applying linear approximations by using derivative rules to simplify functions around specific points for easier calculations.
• Example: Approximating $f\left(x\right)=\sqrt{x+1}$ around $x=3$ using the linear approximation $L\left(x\right)=f\left(3\right)+{f}^{\mathrm{\prime }}\left(3\right)\left(x-3\right)$.
• Example: Approximating $f\left(x\right)=\sqrt{x}$ at $x=4$ using the tangent line equation: $L\left(x\right)=f\left(a\right)+{f}^{\mathrm{\prime }}\left(a\right)\left(x-a\right)$ where $a=4$ and ${f}^{\mathrm{\prime }}\left(x\right)=\frac{1}{2\sqrt{x}}$ yields $L\left(x\right)=2+\frac{1}{4}\left(x-4\right)$.
2. Truncation and Rounding:
• Simplifying complex expressions by truncating higher-order terms or rounding terms involving derivatives for computational ease.
• Example: Approximating $f\left(x\right)={e}^{3x}+\mathrm{ln}\left(2x\right)$ by neglecting terms involving higher-order derivatives beyond a certain point.

• Using the second derivative to construct a quadratic approximation around a point for better accuracy than linear approximations.
• Example: Approximating $f\left(x\right)=\mathrm{ln}\left(x\right)$ near $x=1$ using a quadratic approximation: $Q\left(x\right)=f\left(a\right)+{f}^{\mathrm{\prime }}\left(a\right)\left(x-a\right)+\frac{{f}^{\mathrm{\prime }\mathrm{\prime }}\left(a\right)}{2!}\left(x-a{\right)}^{2}$ where $a=1$ yields $Q\left(x\right)=\left(x-1\right)-\frac{1}{2}\left(x-1{\right)}^{2}$.

### Example:

Let's consider the function $f\left(x\right)={e}^{2x}+\mathrm{cos}\left(3x\right)$. We want to find an approximation of the derivative of $f\left(x\right)$ at $x=0.1$ using the linear approximation.

1. Derivative Calculation:

• Calculate the derivative of $f\left(x\right)$ using the rules of differentiation: ${f}^{\mathrm{\prime }}\left(x\right)=2{e}^{2x}-3\mathrm{sin}\left(3x\right)$.
2. Linear Approximation:

• Evaluate ${f}^{\mathrm{\prime }}\left(0.1\right)$ using the derivative obtained earlier: ${f}^{\mathrm{\prime }}\left(0.1\right)=2{e}^{2\left(0.1\right)}-3\mathrm{sin}\left(3\left(0.1\right)\right)$.
• This gives an exact value for ${f}^{\mathrm{\prime }}\left(0.1\right)$.
3. Using Approximations:

• To approximate ${f}^{\mathrm{\prime }}\left(0.1\right)$ without using exact calculations, one could employ linear approximation if exact calculations are challenging due to complexity or computational limitations.