Derivatives: Power and Constant Multiple Rules

Derivatives of Functions

• The derivative of any constant is zero.
• A derivative is a function providing the slope at a given x-value.
• If f(x) = 8, its graph is a straight line at y = 8, having a slope of zero.
• The derivative of f(x), denoted as f'(x), equals zero in this case.
• The notation d/dx signifies differentiation with respect to x.

Power Rule

• The power rule helps find the derivative of monomials.
• Formula: d/dx (x^n) = n * x^(n-1), where n is a constant.
• Example: The derivative of x^2 using the power rule is 2 * x^(2-1) = 2x.
• The derivative of x^3 is 3x^2
• The derivative of x^4 is 4x^3
• The derivative of x^5 is 5x^4

Constant Multiple Rule

• The constant multiple rule helps find the derivative of a constant times a function.
• Formula: d/dx [c * f(x)] = c * f'(x), where c is a constant
• For 4x^7, bring the 4 to the front: 4 * d/dx (x^7) = 4 * 7x^6 = 28x^6
• The derivative of 8x^4 is 32x^3
• The derivative of 5x^6 is 30x^5
• The derivative of 9x^5 is 45x^4
• The derivative of 6x^7 is 42x^6

Definition of the Derivative

• An alternative way to find derivatives is the definition of the derivative.
• Formula: f'(x) = lim (h->0) [f(x+h) - f(x)] / h
• To find f(x+h), replace x with (x+h) in the original function.
• For f(x) = x^2, f(x+h) = (x+h)^2
• Expanding (x+h)^2 gives x^2 + 2xh + h^2
• Plugging into the formula: lim (h->0) [(x^2 + 2xh + h^2) - x^2] / h
• Simplify: lim (h->0) [2xh + h^2] / h
• Factor out h: lim (h->0) h(2x + h) / h
• Cancel h: lim (h->0) (2x + h)
• As h approaches zero, the result is 2x, confirming the power rule earlier.

Tangent and Secant Lines

• The derivative gives the slope of a tangent line at a specific x-value.
• A tangent line touches the curve at only one point.
• Example: For f(x) = x^2, f'(x) = 2x
• The slope of the tangent line at x = 1 is f'(1) = 2 * 1 = 2
• A secant line intersects the curve at two points.
• The slope of a secant line is calculated using two points: (y2 - y1) / (x2 - x1)
• As the two points of a secant line approach each other, the slope of the secant line approaches the slope of the tangent line.
• For f(x) = x^3, f'(x) = 3x^2
• The slope of the tangent line at x = 2 is f'(2) = 3 * 2^2 = 12

Polynomial functions

• Differentiate each monomial separately using the power rule.
• The derivative of x^3 + 7x^2 - 8x + 6 is 3x^2 + 14x - 8
• f'(x) of 4x^5 + 3x^4 + 9x - 7 is 20x^4 + 12x^3 + 9
• To find the slope of the tangent line at x = 2, plug in 2 into f'(x):
  • If f(x) = 2x^5 + 5x^3 + 3x^2 + 4, then f'(x) = 10x^4 + 15x^2 + 6x
  • f'(2) = 10(16) + 15(4) + 6(2) = 160 + 60 + 12 = 232

Rational Functions

• Rewrite the reciprocal as a negative power of x
• If f(x) = 1/x = x^(-1), rewrite the function as X = -1
• f'(x) = -1 * x^(-2) = -1 / x^2
  • If f(x) = 1 / x^2, then f(x) = x^(-2)
  • f'(x) = -2 * x^(-3) = -2 / x^3
    • Rewrite the function as: f(x) = 8x^(-4)
    • Then we can determine: f'(x) = -32x^(-5) = -32 / x^5
      • Rewrite the function as: f(x) = x^(1/2)
      • The derivative is f'(x) = (1/2) * x^(-1/2) = 1 / (2 * sqrt(x))

Radical Functions

• Rewrite with rational exponents and then use power rule:
• If f(x) = cuberoot(x^(5)) , then rewrite the function as x^(5-3)
• f'(x) = (5/3) * x^(2/3) = (5 * cuberoot(x^2)) / 3
  • If f(x) = 7throot(x^(4)), then rewrite the function as x^(4/7)
  • f'(x) = (4/7) * x^(-3/7) = 4 / (7 * 7throot(x^3))

Combining rules examples

• Distribute to begin:
• If: f(x) = x^2 * (x^3 + 7), find derivative of f(x) = x^5 + 7x^2
• Therefore is: f'(x) = 5x^4 + 14x
  • Begin by expanding:
  • f(x) = (2x - 3)^2=(2x - 3) * (2x - 3) = 4x^2 - 12x + 9 Then derivative is: f'(x) = 8x - 12
    • Simplify:
    • If f(x) = (x^5 + 6x^4 + 5x^3) / x^2 =x^3 + 6x^2 + 5x then f'(x) = 3x^2 + 12x + 5

Trigonometric functions

• The derivative of sin(x) is cos(x)
• The derivative of cos(x) is -sin(x)
• The derivative of sec(x) is sec(x)tan(x)
• The derivative of csc(x) is -csc(x)cot(x)
• The derivative of tan(x) is sec^2(x)
• The derivative of cot(x) is -csc^2(x)

The Product Rule

• Used when finding the derivative of the product of two functions.
• Formula: (f * g)' = f' * g + f * g'
• For x^2 * sin(x), let f = x^2 and g = sin(x)
• Then, f' = 2x and g' = cos(x)
  • Thus: f' * g + f * g'= 2x * sin(x) + x^2 * cos(x)
    • f = is 3x^4 + 7 and g is x^3 - 5x
    • f' = 12x^3 and g' = 3x^2 - 5.
    • Result: is 12x^3(x^3 - 5x) + (3x^4 + 7)(3x^2 - 5)

Challenge Product Rule Example

• Can find the derivative of multiple functions
• (f * g * h)' = f' * g * h + f * g' * h + f * g * h'
• For: f * g* h = (x^3)(tan(x))(3x^2 - 9), the result is: (3x^2) * tan(x) * (3x^2 - 9) + (x^3) * (sec^2(x)) * (3x^2 - 9) + (x^3) * (tan(x)) * (6x)

The Quotient Rule

• Used when finding the derivative of the quotient of two functions.
• (f/g)' = (gf' - fg') / g^2
  • If: f = 5x + 6 , g = 3x - 7
    • Therefore: f' = 5 , g' = 3 - 7
    • Result: is ((3x - 7) * 5 - (5x + 6) * 3)
    • Which can be simplified to ≈ -53 / (3x - 7)^2

Learn about derivatives of functions, including the derivative of a constant and the power rule. Also, learn how to find the derivative of a constant times a function using the constant multiple rule. Examples are provided.