12-02: Derivatives

IROC: a derivative

In essence, the steps to differentiating (from above)

Power Rule

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

Subtract 1 from the exponent

If there is a coefficient, then multiply the dropped value by the coefficient

If you have an exponent of 0, the term is 1 (anything to the power of 0 is 1)

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

Determining equations from equations

Find tangent point (if given x value use that to find y by subbing into the original equation)

Find the derivative

Sub x value into the derivative to find the slope of the tangent

Use y=mx+b, plug in (x,y) from tangent point and slope - isolate for b

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)

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

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)

The chain rule:

f’(x) = g’(h(x)) • h’(x)

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

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

On a graph:

Slope of f(x) | Derivative |
---|---|

+ | is +, above x axis |

0 | is an x intercept |

- | is -, below x axis |

Slope of f’(x) | Derivative |
---|---|

+ | is +, above x axis |

0 | is an x intercept |

- | is -, below x axis |

Displacement | Velocity | Acceleration | |
---|---|---|---|

| Distance and direction that an object has moved from an origin over a period of time | rate of change of displacement of an object with respect to time | rate of change of velocity with respect to time |

| s(t) | s’(t) | s”(t) |

| m | m/s | m/s² |

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

__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

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

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

IROC: a derivative

In essence, the steps to differentiating (from above)

Power Rule

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

Subtract 1 from the exponent

If there is a coefficient, then multiply the dropped value by the coefficient

If you have an exponent of 0, the term is 1 (anything to the power of 0 is 1)

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

Determining equations from equations

Find tangent point (if given x value use that to find y by subbing into the original equation)

Find the derivative

Sub x value into the derivative to find the slope of the tangent

Use y=mx+b, plug in (x,y) from tangent point and slope - isolate for b

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)

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

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)

The chain rule:

f’(x) = g’(h(x)) • h’(x)

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

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

On a graph:

Slope of f(x) | Derivative |
---|---|

+ | is +, above x axis |

0 | is an x intercept |

- | is -, below x axis |

Slope of f’(x) | Derivative |
---|---|

+ | is +, above x axis |

0 | is an x intercept |

- | is -, below x axis |

Displacement | Velocity | Acceleration | |
---|---|---|---|

| Distance and direction that an object has moved from an origin over a period of time | rate of change of displacement of an object with respect to time | rate of change of velocity with respect to time |

| s(t) | s’(t) | s”(t) |

| m | m/s | m/s² |

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

__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

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

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