differentiation

Introduction to Differentiation Rules

• Constant Rule: When differentiating a constant (a number without any variables), the derivative is 0.

• Power Rule: If you have a function of the form $f(x) = x^n$, the derivative is given by:

  • $f'(x) = n imes x^{n-1}$.

  • Example: For $n = 22$, $f(x) = x^{22}$, the derivative is $f'(x) = 22 imes x^{21}$.

General Form for Differentiation

• It is essential that the function is expressed in terms of a variable raised to an exponent.

• Functions that need transformation to have the variable in the exponent should be restructured appropriately.

  • Incorrect Example: $1/x$ is not in the form necessary for direct application of the power rule as it should be $x^{-1}$.

Steps to Differentiate Functions

• Step 1: Ensure the function is written in the appropriate form (i.e. variable with an exponent).

• Step 2: Apply the power rule or other differentiation rules appropriately after rewriting the function.

Differentiating Compound Functions

• In cases where a constant multiplies the function, such as $c imes f(x)$, the derivative is:

  • $c imes f'(x)$.

• If differentiating terms where there are no variables (i.e., constants), the result will be 0.

Example of Differentiation of a Simple Function

• Given a function $f(t) = 4 imes an(t^{0.5})$:

  • Stripping out the constant: $4$.

  • Differentiate the remaining function: $ext{Use the power rule: } f'(t) = 4 imes 0.5 imes t^{-0.5}$ results in multiplication between constants.

Step-by-step Differentiation Examples

• First Example:

  • Let $y = rac{3 x^2}{5}$.

  • Rewrite as $y = rac{3}{5} x^2$.

  • Differentiate:

  • Constants factored out give $rac{3}{5} imes 2x = rac{6x}{5}$.

• Second Example:

  • For $g(x) = x^2 + 4x - 3$,

  • Differentiate to obtain:

  • $g'(x) = 2x + 4$.

Simplifying Expressions

• When simplifying complex expressions:

  • Combine like terms.

  • Example: If $y = rac{x^2 + 4x - 3}{x}$, simplify by dividing each term by $x$, resulting in:

  • $y = x + 4 - rac{3}{x}$.

Introduction to Tangent Line and Derivatives

• Equation of the Tangent Line: For a function $f(x)$, the equation of the tangent line at a specific point can be found using:

  • $y - f(a) = f'(a)(x - a)$,

  • Where $a$ is the x-coordinate of the point of tangency.

Understanding Limits in Calculus

• Concept of Limits:

  • The derivative can be defined using limits:

  • Formula: $ext{lim}_{ riangle x o 0} rac{ riangle y}{ riangle x}$ provides the slope of the tangent line at a point.

Applications of Derivatives in Economics

• The derivative provides insights into optimization problems, such as analyzing profit functions:

  • If $P(x)$ is the profit function, $R(x)$ the revenue function, and $C(x)$ the cost function:

  • $P(x) = R(x) - C(x)$.

• Marginal Analysis:

  • Marginal cost and revenue relate to how changes in production impact overall profitability.

Deriving Cost and Revenue Functions

• Given cost $C(x) = 100x + 200000$, differentiate to find marginal cost:

  • $C'(x) = 100$.

• For revenue, if given demand equation $p = 0.002x + 400$,

  • Rewrite revenue as $R(x) = x imes p$ leading to ultimate derivation.

Conclusion

• Understanding exponential forms, limits, and derivative applications provide foundational tools for more advanced calculus, particularly in real-world applications like economics and optimization problems.