1.2 Differentiation of algebraic functions

Differentiation of algebraic functions is a key concept in calculus. It's all about finding rates of change and slopes of curves. This skill is super useful for solving real-world problems in fields like physics, economics, and engineering.

In this section, we'll learn how to find derivatives using various rules. We'll cover the power rule, product rule, and quotient rule. These tools will help us tackle more complex functions and solve practical problems involving rates of change.

Derivative Definition and Interpretation

Definition and Notation

• The derivative of a function $f(x)$ at a point $x$ is the instantaneous rate of change of the function at that point, which is the slope of the tangent line to the graph of $f(x)$ at the point $(x, f(x))$.
• The derivative of a function $f(x)$ is a new function, denoted by $f'(x)$ or $\frac{dy}{dx}$, that gives the rate of change of $f(x)$ at any point $x$ in the domain of $f$.
• The process of finding the derivative is called differentiation, and a function is differentiable at a point if its derivative exists at that point.

Interpretation and Applications

• The derivative can be interpreted as the rate of change, slope, or velocity in various contexts, such as:
  • The speed of an object at a given time (physics)
  • The rate of change of cost with respect to quantity produced (economics)
• Derivatives have numerous applications in fields like physics, engineering, economics, and biology, where understanding rates of change is crucial for modeling and problem-solving.

Differentiation Rules for Algebraic Functions

Basic Rules

• The constant rule states that the derivative of a constant function is always 0.
  • Example: If $f(x) = 5$, then $f'(x) = 0$.
• The constant multiple rule states that if $c$ is a constant and $f(x)$ is a differentiable function, then the derivative of $cf(x)$ is $c$ times the derivative of $f(x)$: $(cf(x))' = cf'(x)$.
  • Example: If $f(x) = 3x^2$, then $f'(x) = 3 \cdot 2x = 6x$.

Sum and Difference Rules

• The sum rule states that if $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their sum is the sum of their derivatives: $(f(x) + g(x))' = f'(x) + g'(x)$.
  • Example: If $f(x) = x^2 + \sin(x)$, then $f'(x) = 2x + \cos(x)$.
• The difference rule states that if $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their difference is the difference of their derivatives: $(f(x) - g(x))' = f'(x) - g'(x)$.
  • Example: If $f(x) = x^3 - e^x$, then $f'(x) = 3x^2 - e^x$.

Power, Product, and Quotient Rules

Power Rule

• The power rule states that for a function $f(x) = x^n$, where $n$ is a real number, the derivative is $f'(x) = nx^{n-1}$.
  • Example: If $f(x) = x^5$, then $f'(x) = 5x^4$.
  • Example: If $f(x) = \sqrt{x}$, then $f'(x) = \frac{1}{2\sqrt{x}}$.

Product Rule

• The product rule states that if $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their product is: $(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$.
  • Example: If $f(x) = (x^2 + 1)(x^3 - 2)$, then $f'(x) = (2x)(x^3 - 2) + (x^2 + 1)(3x^2)$.
• The product rule is useful when differentiating the product of two or more functions, as it breaks down the derivative into manageable parts.

Quotient Rule

• The quotient rule states that if $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their quotient is: $(\frac{f(x)}{g(x)})' = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$, provided $g(x) \neq 0$.
  • Example: If $f(x) = \frac{x^2 + 1}{x - 2}$, then $f'(x) = \frac{(x-2)(2x) - (x^2+1)(1)}{(x-2)^2}$.
• The quotient rule is used when differentiating the ratio of two functions, and it accounts for the derivatives of both the numerator and denominator.

Higher-Order Derivatives

Definition and Notation

• The second derivative of a function $f(x)$, denoted by $f''(x)$ or $\frac{d^2y}{dx^2}$, is the derivative of the first derivative, $f'(x)$.
  • Example: If $f(x) = x^3$, then $f'(x) = 3x^2$ and $f''(x) = 6x$.
• The third derivative of a function $f(x)$, denoted by $f'''(x)$ or $\frac{d^3y}{dx^3}$, is the derivative of the second derivative, $f''(x)$.
  • Example: If $f(x) = x^4$, then $f'(x) = 4x^3$, $f''(x) = 12x^2$, and $f'''(x) = 24x$.

Finding Higher-Order Derivatives

• Higher-order derivatives are found by repeatedly applying differentiation rules to the previous derivative.
• The nth derivative of a function $f(x)$ is denoted by $f^{(n)}(x)$ or $\frac{d^ny}{dx^n}$.
  • Example: If $f(x) = e^x$, then $f'(x) = e^x$, $f''(x) = e^x$, and $f^{(n)}(x) = e^x$ for all $n$.
• Higher-order derivatives are used in various applications, such as analyzing the concavity of functions, solving differential equations, and approximating functions using Taylor series.

Differentiation Applications

Optimization Problems

• Differentiation can be used to solve optimization problems, such as finding the maximum or minimum values of a function in a given context.
  • Example: A manufacturer wants to minimize the cost of producing a certain product, given constraints on materials and labor. Differentiation can be used to find the optimal production quantity that minimizes the total cost function.
• Optimization problems often involve setting the derivative of a function equal to zero and solving for the critical points, then evaluating the function at these points to determine the maximum or minimum values.

Related Rates

• Differentiation can be applied to solve related rates problems, where the rates of change of two or more related quantities are given, and the rate of change of one quantity is to be determined.
  • Example: If a balloon is being inflated and its radius is increasing at a rate of 2 cm/s, differentiation can be used to find the rate at which the volume of the balloon is increasing at a specific moment.
• Related rates problems require identifying the relationship between the quantities, differentiating both sides of the equation with respect to time, and solving for the desired rate of change.

Motion Analysis

• Differentiation can be used to analyze the motion of objects, such as finding the velocity and acceleration of an object given its position function.
  • Example: If an object's position is given by the function $s(t) = t^3 - 2t^2 + 4t$, where $s$ is in meters and $t$ is in seconds, differentiation can be used to find the object's velocity and acceleration at any time $t$.
• The first derivative of a position function gives the velocity, while the second derivative gives the acceleration. These concepts are fundamental in physics and engineering applications involving motion.

Economics Applications

• Marginal analysis in economics uses derivatives to determine the rate of change in cost, revenue, or profit with respect to the number of units produced or sold.
  • Example: If the total cost function for producing $x$ units of a product is given by $C(x) = 500 + 10x + 0.2x^2$, differentiation can be used to find the marginal cost, which is the additional cost of producing one more unit.
• Derivatives are also used in economics to analyze the elasticity of demand, optimize production levels, and make informed decisions based on marginal costs and benefits.