Derivatives (1)

Understanding Derivatives

• Definition of a Derivative: A derivative is a function that provides the slope of a given function at a particular x-value.

  • Example: If (f(x) = 8), the graph is a horizontal line at (y = 8), where the slope is (0). Therefore, (f' = 0).

Constant Derivatives

• The derivative of a constant is always 0.

  • Example: (f'(5) = 0) and (f'(-7) = 0)

Derivatives of Monomials using the Power Rule

• Power Rule: The derivative of (x^n) is (n , x^{n-1}).

  • Example 1: (f(x) = x^2) implies (f' = 2x^{2-1} = 2x)

  • Example 2: (f(x) = x^3) thus (f' = 3x^{3-1} = 3x^2)

  • Example 3: (f(x) = x^4) yields (f' = 4x^{4-1} = 4x^3)

  • Example 4: (f(x) = x^5) results in (f' = 5x^{5-1} = 5x^4)

Finding the Derivative of Monomials with Constants

• Constant Multiple Rule: If (f(x) = c , g(x)) then (f' = c , g').

  • Example: To find the derivative of (f(x) = 4x^7):

    • Identify (c = 4) and (g(x) = x^7).

    • The derivative results in: (f' = 4(7x^{7-1}) = 28x^6).

Practice Derivatives of Monomials

• Find the derivatives of:

  • (f(x) = 8x^4) → (f' = 32x^3)

  • (f(x) = 5x^6) → (f' = 30x^5)

  • (f(x) = 9x^5) → (f' = 45x^4)

  • (f(x) = 6x^7) → (f' = 42x^6)

Definition of the Derivative via Limits

• The derivative can be calculated using the limit definition: [ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

  • Example for (f(x) = x^2):

    • Substitute: (f(x+h) = (x+h)^2 = x^2 + 2xh + h^2)

    • Leads to: (f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} = 2x)

Slope of Tangent Lines

• The derivative can be evaluated at a specific x value to find the slope of the tangent line.

  • For (f'(x) = 2x) at (x = 1): (f'(1) = 2), hence slope = 2.

Tangent vs Secant Lines

• Tangent Line: Touches the curve at one point.

• Secant Line: Touches the curve at two points. (m_{secant} = \frac{y_2 - y_1}{x_2 - x_1})

  • As two points converge, the secant slope approaches the tangent slope.

Example of Calculating Slope of Tangent Lines

• Finding the slope for (f(x) = x^3) at (x = 2):

  • (f'(x) = 3x^2) leads to (f'(2) = 3(2^2) = 12).

• To approximate the tangent using secant:

  1. Choose values around (2) (e.g., (1.9) and (2.1)).

  2. Calculate slopes.

Derivative of Polynomial Functions

• Example: For (f(x) = x^3 + 7x^2 - 8x + 6), find (f'):

  • Therefore, (f' = 3x^2 + 14x - 8).

Complex Functions and Chain Rule

• Use the Chain Rule for derivatives of composite functions.

Derivatives of Trigonometric Functions

• Important derivatives to memorize:

  • (\sin{x} \rightarrow \cos{x})

  • (\cos{x} \rightarrow -\sin{x})

  • (\tan{x} \rightarrow \sec^2{x})

  • (\sec{x} \rightarrow \sec{x} \tan{x})

Product Rule for Derivatives

• For products of two functions:

  • (d(uv)/dx = u'v + uv') where (u = x^2, v = \sin{x}).

Quotient Rule for Derivatives

• For quotients, use

  • (d(f/g)/dx = \frac{g f' - f g'}{g^2})

• Example: (f(x) = \frac{5x+6}{3x-7})

Recap

• A derivative is a powerful tool to find slopes and understand the behavior of functions.

• Mastery of rules like the Power Rule, Product Rule, Quotient Rule, and the ability to use limits is essential for calculus.