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