Derivatives - Examples of Chain Rule

• The chain rule is used to calculate the derivative of composite functions.
• It allows us to find the derivative of a function that is composed of two or more functions.
• The chain rule states that the derivative of the composition of two functions is the derivative of the outer function multiplied by the derivative of the inner function. Example:
• If f(x) = (x^2 + 1)^3, find f’(x). Solution:
• Let’s consider g(x) = x^3 and h(x) = x^2 + 1.
• Applying the chain rule, we have f’(x) = g’(h(x)) * h’(x).
• Differentiating g(x), we get g’(x) = 3x^2.
• Differentiating h(x), we get h’(x) = 2x.
• Substituting g’(x) and h’(x) in the chain rule formula, we have f’(x) = 3(x^2 + 1)^2 * 2x.
• Simplifying further, f’(x) = 6x(x^2 + 1)^2. Equation:
• The equation for the chain rule is:
  • f’(x) = g’(h(x)) * h’(x)

Derivatives - Product Rule

• The product rule is used to find the derivative of a product of two functions.
• It is used when we have a function that can be expressed as a product of two functions. Example:
• If f(x) = x^2 * sin(x), find f’(x). Solution:
• We have two functions here: g(x) = x^2 and h(x) = sin(x).
• Applying the product rule, we get f’(x) = g’(x) * h(x) + g(x) * h’(x).
• Differentiating g(x), we get g’(x) = 2x.
• Differentiating h(x), we get h’(x) = cos(x).
• Substituting g’(x) and h’(x) in the product rule formula, we have f’(x) = 2x * sin(x) + x^2 * cos(x). Equation:
• The equation for the product rule is:
  • f’(x) = g’(x) * h(x) + g(x) * h’(x)

Derivatives - Quotient Rule

• The quotient rule is used to find the derivative of a quotient of two functions.
• It is used when we have a function that can be expressed as the ratio of two functions. Example:
• If f(x) = x^3 / cos(x), find f’(x). Solution:
• We have two functions here: g(x) = x^3 and h(x) = cos(x).
• Applying the quotient rule, we get f’(x) = (g’(x) * h(x) - g(x) * h’(x)) / (h(x))^2.
• Differentiating g(x), we get g’(x) = 3x^2.
• Differentiating h(x), we get h’(x) = -sin(x).
• Substituting g’(x) and h’(x) in the quotient rule formula, we have f’(x) = (3x^2 * cos(x) - x^3 * (-sin(x))) / (cos(x))^2.
• Simplifying further, f’(x) = (3x^2 * cos(x) + x^3 * sin(x)) / cos^2(x). Equation:
• The equation for the quotient rule is:
  • f’(x) = (g’(x) * h(x) - g(x) * h’(x)) / (h(x))^2

Derivatives - Logarithmic Functions

• The derivative of logarithmic functions can be found using differentiation rules. Example:
• If f(x) = ln(x^2 + 3), find f’(x). Solution:
• Using the chain rule, we have f’(x) = (1 / (x^2 + 3)) * (2x).
• Simplifying further, f’(x) = (2x) / (x^2 + 3). Equation:
• The equation for the derivative of logarithmic functions is:
  • f’(x) = (1 / (x)) * (g’(x)) Where g(x) is the function inside the logarithm.

Derivatives - Exponential Functions

• The derivative of exponential functions can be found using differentiation rules. Example:
• If f(x) = e^(2x), find f’(x). Solution:
• Using the chain rule, we have f’(x) = e^(2x) * (2).
• Simplifying further, f’(x) = 2e^(2x). Equation:
• The equation for the derivative of exponential functions is:
  • f’(x) = k * e^(kx) Where k is a constant.

Derivatives - Trigonometric Functions

• The derivative of trigonometric functions can be found using differentiation rules. Example:
• If f(x) = sin(3x), find f’(x). Solution:
• Using the chain rule, we have f’(x) = 3cos(3x). Equation:
• The equation for the derivative of trigonometric functions is:
  • f’(x) = k * cos(kx) Where k is a constant.

11. Derivatives - Examples of Chain Rule

• The chain rule is used to calculate the derivative of composite functions.
• It allows us to find the derivative of a function that is composed of two or more functions.
• The chain rule states that the derivative of the composition of two functions is the derivative of the outer function multiplied by the derivative of the inner function. Example:
• If f(x) = (x^2 + 1)^3, find f’(x).
• Let’s consider g(x) = x^3 and h(x) = x^2 + 1.
• Applying the chain rule, we have f’(x) = g’(h(x)) * h’(x).
• Differentiating g(x), we get g’(x) = 3x^2.
• Differentiating h(x), we get h’(x) = 2x.
• Substituting g’(x) and h’(x) in the chain rule formula, we have f’(x) = 3(x^2 + 1)^2 * 2x.
• Simplifying further, f’(x) = 6x(x^2 + 1)^2. Equation:
• The equation for the chain rule is:
  • f’(x) = g’(h(x)) * h’(x)

12. Derivatives - Product Rule

• The product rule is used to find the derivative of a product of two functions.
• It is used when we have a function that can be expressed as a product of two functions. Example:
• If f(x) = x^2 * sin(x), find f’(x).
• We have two functions here: g(x) = x^2 and h(x) = sin(x).
• Applying the product rule, we get f’(x) = g’(x) * h(x) + g(x) * h’(x).
• Differentiating g(x), we get g’(x) = 2x.
• Differentiating h(x), we get h’(x) = cos(x).
• Substituting g’(x) and h’(x) in the product rule formula, we have f’(x) = 2x * sin(x) + x^2 * cos(x). Equation:
• The equation for the product rule is:
  • f’(x) = g’(x) * h(x) + g(x) * h’(x)

13. Derivatives - Quotient Rule

• The quotient rule is used to find the derivative of a quotient of two functions.
• It is used when we have a function that can be expressed as the ratio of two functions. Example:
• If f(x) = x^3 / cos(x), find f’(x).
• We have two functions here: g(x) = x^3 and h(x) = cos(x).
• Applying the quotient rule, we get f’(x) = (g’(x) * h(x) - g(x) * h’(x)) / (h(x))^2.
• Differentiating g(x), we get g’(x) = 3x^2.
• Differentiating h(x), we get h’(x) = -sin(x).
• Substituting g’(x) and h’(x) in the quotient rule formula, we have f’(x) = (3x^2 * cos(x) - x^3 * (-sin(x))) / (cos(x))^2.
• Simplifying further, f’(x) = (3x^2 * cos(x) + x^3 * sin(x)) / cos^2(x). Equation:
• The equation for the quotient rule is:
  • f’(x) = (g’(x) * h(x) - g(x) * h’(x)) / (h(x))^2

14. Derivatives - Logarithmic Functions

• The derivative of logarithmic functions can be found using differentiation rules. Example:
• If f(x) = ln(x^2 + 3), find f’(x).
• Using the chain rule, we have f’(x) = (1 / (x^2 + 3)) * (2x).
• Simplifying further, f’(x) = (2x) / (x^2 + 3). Equation:
• The equation for the derivative of logarithmic functions is:
  • f’(x) = (1 / (x)) * (g’(x)) Where g(x) is the function inside the logarithm.

15. Derivatives - Exponential Functions

• The derivative of exponential functions can be found using differentiation rules. Example:
• If f(x) = e^(2x), find f’(x).
• Using the chain rule, we have f’(x) = e^(2x) * (2).
• Simplifying further, f’(x) = 2e^(2x). Equation:
• The equation for the derivative of exponential functions is:
  • f’(x) = k * e^(kx) Where k is a constant.

16. Derivatives - Trigonometric Functions

• The derivative of trigonometric functions can be found using differentiation rules. Example:
• If f(x) = sin(3x), find f’(x).
• Using the chain rule, we have f’(x) = 3cos(3x). Equation:
• The equation for the derivative of trigonometric functions is:
  • f’(x) = k * cos(kx) Where k is a constant.

17. Derivatives - Inverse Trigonometric Functions

• The derivative of inverse trigonometric functions can be found using differentiation rules. Example:
• If f(x) = arcsin(2x), find f’(x).
• Using the chain rule, we have f’(x) = 2 / sqrt(1 - (2x)^2). Equation:
• The equation for the derivative of inverse trigonometric functions is:
  • f’(x) = 1 / sqrt(1 - (g(x))^2) * g’(x) Where g(x) is the function inside the inverse trigonometric function.

18. Derivatives - Hyperbolic Functions

• The derivative of hyperbolic functions can be found using differentiation rules. Example:
• If f(x) = sinh(3x), find f’(x).
• Using the chain rule, we have f’(x) = 3cosh(3x). Equation:
• The equation for the derivative of hyperbolic functions is:
  • f’(x) = k * cosh(kx) Where k is a constant.

19. Derivatives - Implicit Differentiation

• Implicit differentiation is used to find the derivative of an equation where y is not expressed in terms of x explicitly. Example:
• If we have the equation x^2 + y^2 = 9, find dy/dx.
• Differentiating both sides of the equation, we get 2x + 2yy’ = 0.
• Solving for y’, we have y’ = -x/y. Equation:
• The equation for implicit differentiation is:
  • Differentiate both sides of the equation with respect to x and solve for the derivative of y.

20. Derivatives - Higher Order Derivatives

• Higher order derivatives represent the derivative of a function with respect to the same variable multiple times. Example:
• If f(x) = 3x^4 - 5x^2 + 2, find f’’(x).
• Differentiating f(x) twice, we get f’’(x) = 72x^2 - 10. Equation:
• The equation for higher order derivatives is:
  • The derivative of nth order of a function f(x) is denoted as f^(n)(x).

21. Derivatives - Optimization Problems

• Optimization problems involve finding the maximum or minimum values of a function.
• The derivative can be used to solve these problems by identifying critical points and using the first or second derivative test. Example:
• A rectangular garden is to be fenced with 100 meters of fencing. Find the dimensions that maximize the area of the garden.
• Let’s assume the length of the garden is L and the width is W.
• We know that the perimeter of the garden is 2L + 2W = 100.
• Solving this equation for L, we get L = 50 - W.
• The area of the garden is given by A = LW.
• Substituting the value of L in terms of W, we have A = (50 - W)W.
• To find the maximum area, we differentiate A with respect to W and set it equal to zero.
• Differentiating A, we get A’ = 50 - 2W.
• Setting A’ = 0, we have 50 - 2W = 0.
• Solving for W, we get W = 25.
• Substituting W = 25 in the equation for L, we get L = 50 - 25 = 25.
• Therefore, the dimensions that maximize the area are L = 25 meters and W = 25 meters. Equation:
• The equation for optimization problems is to differentiate the function and set it equal to zero to find critical points.

22. Derivatives - Related Rates Problems

• Related rates problems involve finding the rate of change of one quantity with respect to another.
• The derivative can be used to solve these problems by relating the variables and differentiating the equation with respect to time. Example:
• Two cars are driving towards each other on a straight road. The first car is traveling at 60 km/h and the second car is traveling at 70 km/h. At what rate are the two cars approaching each other when they are 100 km apart?
• Let’s assume the distance between the two cars is D, and the rate at which the distance is changing is dD/dt.
• Applying the Pythagorean theorem, we have D^2 = x^2 + y^2, where x and y are the distances traveled by the first and second cars, respectively.
• Differentiating both sides of the equation with respect to time, we get 2D(dD/dt) = 2x(dx/dt) + 2y(dy/dt).
• Given dx/dt = 60 km/h and dy/dt = 70 km/h, substituting these values in the equation, we have 2(100)(dD/dt) = 2(60)(x) + 2(70)(y).
• Since x + y = D, we can rewrite the equation as 2(100)(dD/dt) = 2(60)(D - y) + 2(70)(y).
• Simplifying further, we have 200(dD/dt) = 120D - 120y + 140y.
• Combining like terms, we get 200(dD/dt) = 120D + 20y.
• Substituting D = 100 km and y = 100 - x = 100 - 60 = 40 km, we have 200(dD/dt) = 120(100) + 20(40).
• Simplifying, we get 200(dD/dt) = 12000 + 800.
• Solving for dD/dt, we have dD/dt = (12000 + 800) / 200 = 64 km/h.
• Therefore, the two cars are approaching each other at a rate of 64 km/h. Equation:
• The equation for related rates problems is to differentiate the equation with respect to time and relate the variables.

23. Derivatives - L’Hospital’s Rule

• L’Hospital’s Rule is used to find the limit of the ratio of two functions when both functions approach zero or infinity.
• It states that if a limit of the form 0/0 or ∞/∞ arises, then the limit of the ratio is equal to the limit of the ratio of their derivatives. Example:
• Find the limit as x approaches 0 of (sin(x) / x).
• Directly substituting 0 for x in the function would result in an indeterminate form (0/0).
• Applying L’Hospital’s Rule, we can differentiate both the numerator and denominator and then evaluate the limit again.
• Differentiating sin(x) with respect to x, we get cos(x).
• Differentiating x with respect to x, we get 1.
• Taking the limit again, we have the limit as x approaches 0 of (cos(x) / 1).
• Evaluating this limit, we get cos(0) / 1 = 1.
• Therefore, the limit of (sin(x) / x) as x approaches 0 is equal to 1. Equation:
• The equation for applying L’Hospital’s Rule is to differentiate the numerator and denominator and evaluate the limit again.

24. Derivatives - Taylor Series

• The Taylor series expansion represents a function as an infinite sum of terms.
• It can be used to approximate the value of a function at a particular point using its derivatives. Example:
• Find the Taylor series expansion for the function f(x) = sin(x) centered at a = 0.
• The general formula for