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B1.1 & B2.1 – Derivatives of Power Functions - Power Rule, Constant Rule, Sum and Difference Rule

B1.1 & B2.1 – Derivatives of Power Functions - Power Rule, Constant Rule, Sum and Difference Rule . IBHL/SL Y2 - Santowski. (A) Review. The equation used to find a tangent line or an instantaneous rate of change is: which we also then called a derivative. So derivatives are calculated as .

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B1.1 & B2.1 – Derivatives of Power Functions - Power Rule, Constant Rule, Sum and Difference Rule

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  1. B1.1 & B2.1 – Derivatives of Power Functions - Power Rule, Constant Rule, Sum and Difference Rule IBHL/SL Y2 - Santowski

  2. (A) Review • The equation used to find a tangent line or an instantaneous rate of change is: • which we also then called a derivative. • So derivatives are calculated as . • Since we can differentiate at any point on a function, we can also differentiate at every point on a function (subject to continuity)  therefore, the derivative can also be understood to be a function itself.

  3. (B) Finding Derivatives Without Using Limits – Differentiation Rules • We will now develop a variety of useful differentiation rules that will allow us to calculate equations of derivative functions much more quickly (compared to using limit calculations each time) • First, we will work with simple power functions • We shall investigate the derivative rules by means of the following algebraic and GC investigation (rather than a purely “algebraic” proof)

  4. (i) f(x) = 3 is called a constant function  graph and see why. What would be the rate of change of this function at x = 6? x = -1, x = a? We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is f `(x) = 0 (C) Constant Functions

  5. (ii) f(x) = 4x or f(x) = -½x or f(x) = mx  linear fcns What would be the rate of change of this function at x = 6? x = -1, x = a. We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is the constant function f `(x) = m (D) Linear Functions

  6. (iii) f(x) = x2 or 3x2 or ax2 quadratics What would be the rate of change of this function at x = 6? x = -1, x = a. We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is the linear function f `(x) = 2ax (E) Quadratic Functions

  7. (iv) f(x) = x3 or ¼ x3 or ax3? What would be the rate of change of this function at x = 6? x = -1, x = a. We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is the quadratic function f `(x) = 3ax2 (F) Cubic Functions

  8. (v) f(x) = x4 or ¼ x4 or ax4? What would be the rate of change of this function at x = 6? x = -1, x = a. We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is the cubic function f `(x) = 4ax3 (G) Quartic Functions

  9. (H) Summary • We can summarize the observations from previous slides  the derivative function is one power less and has the coefficient that is the same as the power of the original function • i.e. x4  4x3 • GENERAL PREDICTION  xn  nxn-1

  10. (I) Algebraic Verification • do a limit calculation on the following functions to find the derivative functions: • (i) f(x) = k • (ii) f(x) = mx • (iii) f(x) = x2 • (iv) f(x) = x3 • (v) f(x) = x4 • (vi) f(x) = xn

  11. (J) Power Rule • We can summarize our findings in a “rule”  we will call it the power rule as it applies to power functions • If f(x) = xn, then the derivative function f `(x) = nxn-1 • Which will hold true for all n • (Realize that in our investigation, we simply “tested” positive integers  we did not test negative numbers, nor did we test fractions, nor roots)

  12. (K) Sum, Difference, Constant Multiple Rules • We have seen derivatives of simple power functions, but what about combinations of these functions? What is true about their derivatives?

  13. (vi) f(x) = x² + 4x - 1 What would be the rate of change of this function at x = 6? x = -1, x = a. We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is the “combined derivative” of f `(x) = 2x + 4 (L) Example #1

  14. (L) Example #1

  15. (vii) f(x) = x4 – 3x3 + x2 -½x + 3 What would be the rate of change of this function at x = 6? x = -1, x = a. We could do a limit calculation to find the derivative value  but we will graph it on the GC and graph its derivative. So the derivative function equation is the “combined derivative” f `(x) = 4x3 – 3x2 + 2x – ½ (M) Example 2

  16. (N) Links • Visual Calculus - Differentiation Formulas • Calculus I (Math 2413) - Derivatives - Differentiation Formulas from Paul Dawkins • Calc101.com Automatic Calculus featuring a Differentiation Calculator • Some on-line questions with hints and solutions

  17. (O) Homework • IBHL/SL Y2 – Stewart, 1989, Chap 2.2, p83, Q7-10 • Stewart, 1989, Chap 2.3, p88, Q1-3eol,6-10

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