12-02: Derivatives

Derivative Rules

IROC: a derivative

In essence, the steps to differentiating (from above)

==Power Rule==

1. Drop the exponent’s value down to the front of the term

2. Subtract 1 from the exponent

   1. If there is a coefficient, then multiply the dropped value by the coefficient
   2. If you have an exponent of 0, the term is 1 (anything to the power of 0 is 1)

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Rational (Radicals) and Negative Exponents:

• To differentiate radicals: express it as a power with a rational exponent
• To differentiate a power of x that is in the denominator: express as a power with a negative exponent

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Determining equations from equations

1. Find tangent point (if given x value use that to find y by subbing into the original equation)
2. Find the derivative
3. Sub x value into the derivative to find the slope of the tangent
4. Use y=mx+b, plug in (x,y) from tangent point and slope - isolate for b

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The Product Rule

Instead of expanding out, there is another way to find the derivative of p(x)=f(x)g(x)

p’(x) = f’(x)g(x) + g’(x)f(x)

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The Quotient Rule

Method used to differentiate functions in the form q(x) = f(x)/g(x)

The Chain Rule

Composite function: one function being subbed into another function

• What’s on the inside of the brackets/radical/etc
• Parent function (what’s on the outside)

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The chain rule:

f’(x) = g’(h(x)) • h’(x)

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• A composite function g(h(x)) consists of an outer function g(x) and an inner function h(x). The chain rule is an effective way of differentiating a composite function by first: differentiating the outer function with respect to the inner function, then multiplying by the derivative of the function

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Velocity, Acceleration, and Second Derivatives

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Second Derivatives

Second derivative: f’’(x) determined by differentiating the first derivative (same process, just starting from the first derivative)

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On a graph:

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Using Derivatives to Analyze the Motion of Objects Travelling in a Straight Line

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Speed and velocity are different

• Speed: a quantity or scale that describes the magnitude of motion but does not describe direction
• Velocity: a vector quantity with both magnitude and direction
  • Answer for velocity questions can be either + or - and the sign indicates the direction at which the object is travelling

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Possible velocity v(t) and acceleration a(t) values:

• when v(t) = 0: object is at rest
• when v(t) > 0: object is moving in a + direction
• when v(t) < 0: object is moving in a - direction
• when a(t) > 0: object is accelerating; velocity is increasing
• when a(t) < 0: object is decelerating; velocity is decreasing

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• if v(t) • a(t) > 0: object is speeding up
• if v(t) • a(t) > 0: object is slowing down

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