Errors and Approximations
Errors in Using Derivative Rules:

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 $f(x)={x}^{4}+3{x}^{2}2$leading to ${f}^{\mathrm{\prime}}(x)=4{x}^{3}+6x$ instead of the correct derivative ${f}^{\mathrm{\prime}}(x)=4{x}^{3}+6x$.

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 $f(x)=\mathrm{sin}(3{x}^{2})$, leading to an incorrect derivative.
Approximations Using Derivative Rules:

Linear Approximation:
 Applying linear approximations by using derivative rules to simplify functions around specific points for easier calculations.
 Example: Approximating $f(x)=\sqrt{x+1}$
 $f(x)=\sqrt{x}$
 Truncation and Rounding:

 Simplifying complex expressions by truncating higherorder terms or rounding terms involving derivatives for computational ease.
 Example: Approximating $f(x)={e}^{3x}+\mathrm{ln}(2x)$ by neglecting terms involving higherorder derivatives beyond a certain point.
Quadratic Approximation (Second Derivative):

 Using the second derivative to construct a quadratic approximation around a point for better accuracy than linear approximations.
 Example: Approximating $f(x)=\mathrm{ln}(x)$ near $x=1$ using a quadratic approximation: $Q(x)=f(a)+{f}^{\mathrm{\prime}}(a)(xa)+\frac{{f}^{\mathrm{\prime}\mathrm{\prime}}(a)}{2!}(xa{)}^{2}$ where $a=1$ yields $Q(x)=(x1)\frac{1}{2}(x1{)}^{2}$.
Example:
Let's consider the function $f(x)={e}^{2x}+\mathrm{cos}(3x)$. We want to find an approximation of the derivative of $f(x)$ at $x=0.1$ using the linear approximation.

Derivative Calculation:
 Calculate the derivative of $f(x)$ using the rules of differentiation: ${f}^{\mathrm{\prime}}(x)=2{e}^{2x}3\mathrm{sin}(3x)$.

Linear Approximation:
 Evaluate ${f}^{\mathrm{\prime}}(0.1)$ using the derivative obtained earlier: ${f}^{\mathrm{\prime}}(0.1)=2{e}^{2(0.1)}3\mathrm{sin}(3(0.1))$.
 This gives an exact value for ${f}^{\mathrm{\prime}}(0.1)$.

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