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Topic 2.1 Extended F – Derivative shortcuts

Instantaneous Velocity. The derivative shortcut. The derivative shortcut. for powers of t. for powers of t. Topic 2.1 Extended F – Derivative shortcuts.  We have two equivalent definitions of instantaneous velocity, both of which are hard to use:.  x  t. d x dt. limit  t →0. =.

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Topic 2.1 Extended F – Derivative shortcuts

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  1. Instantaneous Velocity The derivative shortcut The derivative shortcut for powers of t for powers of t Topic 2.1 ExtendedF – Derivative shortcuts We have two equivalent definitions of instantaneous velocity, both of which are hard to use: x t dx dt limit t→0 = v = x(t + t) - x(t) t dx dt limit t→0 v = = We learned the following shortcut to find the derivative of powers of t: If x = Ctn, then v = nCtn-1. In terms of Leibniz' notation we can write the same rule as: If x = Ctn, then = nCtn-1. dx dt

  2. The derivative shortcut for x = sin t The derivative shortcut The derivative shortcut for x = et for x = cos t Topic 2.1 ExtendedF – Derivative shortcuts Now we have discovered (using the method of slopes) three new shortcuts: We learned the following shortcut to find the derivative of x =sin t: If x = sint, then v = cost. If x = sint, then = cost. OR dx dt We learned the following shortcut to find the derivative of x = cos t: If x = cost, then v = -sint. If x = cost, then = -sint. OR dx dt We learned the following shortcut to find the derivative of x = et: If x = et, then v = et. If x = et, then = et. OR dx dt

  3. dx dt dx dt dx dt = 3et = -4sint = 2cost Topic 2.1 ExtendedF – Derivative shortcuts It turns out that usually our x formulas are not quite so simple. For example, suppose x = 2 sin(3t). Of course, we could graph it and find the formula for v, or we could use one of the definitions of the derivative. For our purposes we will just give you the generalized rules here, where A and B are constants: RULE 1: The derivative of a constant times a function is that same constant times the derivative of the function. Formula Example dx dt x = 2sint If x = Asint, then = Acost. dx dt x = 4cost If x = Acost, then = -Asint. dx dt x = 3et If x = Aet, then = Aet.

  4. dx dt dx dt dx dt = -3sin(3t) = 4e4t = 2cos(2t) Topic 2.1 ExtendedF – Derivative shortcuts The argument of a function is what the sine, or cosine or exponential acts on: x = sin(2t) x = cos(3t) x = e4t RULE 2: The derivative of a function with an IMPERFECT ARGUMENT (one other than just a t) is the derivative of the function times the derivative of the argument. Formula Example dx dt x = sin(2t) If x = sin(Bt), = Bcos(Bt). dx dt x = cos(3t) If x = cos(Bt), = -Bsin(Bt). dx dt x = e4t If x = eBt, then = BeBt. FYI: Note that the ARGUMENT does NOT change. FYI: Rule 2 is also known as the CHAIN RULE.

  5. dx dt dx dt dx dt = = = Topic 2.1 ExtendedF – Derivative shortcuts RULE 3: The derivative of a SUM OF TWO FUNCTIONS is the sum of the derivatives of the two functions. Example cost + -sint) x = sint + cost x = -5t2 + 6t + 2 -10t + 6 + 2cos2t 12e4t x = 3e4t + sin2t FYI: Some of the derivatives need the CHAIN RULE. FYI: Rule 4 is also known as the PRODUCT RULE.

  6. dx dt dx dt dx dt = = = Topic 2.1 ExtendedF – Derivative shortcuts RULE 4: The derivative of a PRODUCT OF TWO FUNCTIONS is the derivative of the first times the original second PLUS the original first times the derivative of the second. Example (cost) ·(cost) +(sint) x = (sint)(cost) ·(-sint) (2cos2t) ·(cost) +(sin2t) x = (sin2t)(cost) ·(-sint) (12e4t) ·(sin2t) +(3e4t) ·(2cos2t) x = (3e4t)(sin2t) FYI: Some of the derivatives need the CHAIN RULE. FYI: Rule 4 is also known as the PRODUCT RULE.

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