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Understanding Differentiation Formulas

Learn differentiation rules such as Constant Rule, Power Rule, Extended Power Rule, Sum Rule, Product Rule, Quotient Rule, and application in finding tangent and normal lines to curves. Discover derivatives of trig functions with practical examples.

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Understanding Differentiation Formulas

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  1. 2.3 • Differentiation Formulas

  2. Differentiation Formulas • Let’s start with the simplest of all functions, the constant function f(x) = c. The graph of this function is the horizontal line y = c, which has slope 0, so we must have f(x) = 0. • The graph of f(x) = c is the line y = c, so f(x) = 0.

  3. Constant Rule • Using the formal definition of derivative:

  4. Power Rule • For functions f(x) = xn, where n is a positive integer:

  5. Proof by formal definition of derivative: • For n = 4 we find the derivative of f(x) = x4 as follows: • (x4) = 4x3

  6. Practice: Find each derivative • (a) If f(x) = x6 • (b) If y = x1000 • (c) If y = t4 • (d) (r3)

  7. Extended Power Rule

  8. PROPERTIES AND RULES OF DERIVATIVES

  9. Constant Multiple Rule

  10. Sum Rule • the derivative of a sum of functions is the sum of the derivatives.

  11. Difference Rule

  12. Example: • (x8 + 12x5 – 4x4 + 10x3 – 6x + 5) • = (x8)+12 (x5)–4 (x4)+10 (x3)–6 (x)+ (5) • = 8x7 + 12(5x4) – 4(4x3) + 10(3x2) – 6(1) + 0 • = 8x7+ 60x4 – 16x3 + 30x2 – 6

  13. Product Rule • the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. Or:

  14. Quotient Rule • the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. • Or: • Or:

  15. Example: • Let . Then

  16. Use quotient rule? • Don’t use the Quotient Rule every time you see a quotient. • Sometimes, when there is only ONE term in the quotient, it’s easier to rewrite the expression as a sum of power terms, then use the power rule. • Example: • f(x) = • We can use the quotient rule but it is much easier to perform the division first and write the function as: • f(x) = 3x + 2x –1 2

  17. Examples: • (a) If y = , then • = –x–2 • = • (b)

  18. Practice: • Differentiate the function f(t) =(a + bt). • Solution 1:Using the Product Rule, we have

  19. Practice – Solution2 • cont’d • If we first use the laws of exponents to rewrite f(t), then we can proceed directly without using the Product Rule.

  20. Normal line at a point: • The differentiation rules enable us to find tangent lines without having to resort to the definition of a derivative. • They also enable us to find normal lines. • The normal line to a curve C at point P is the line throughP that is perpendicular to the tangent line at P.

  21. Example: • Find equations of the tangent line and normal line to the curve • y = (1 + x2) at the point (1, ). • Solution: According to the Quotient Rule, we have

  22. Example – Solution • cont’d • So the slope of the tangent line at (1, ) is • We use the point-slope form to write an equation of the tangent line at (1, ): • y – = – (x – 1) or y =

  23. Example – Solution • cont’d • The slope of the normal line at (1, ) is the negative reciprocal of , namely 4, so an equation is • y – = 4(x – 1) or y = 4x – • The curve and its tangent and normal lines are graphed in Figure 5. • Figure 5

  24. Summary of Rules

  25. 2.4 • Derivatives of Trig Functions

  26. Example 1 • Differentiate y = x2 sin x. • Solution: • Using the Product Rule

  27. Example 2 • An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0 (note that the downward direction is positive.) • Its position at time t is • s = f (t) = 4 cos t • Find the velocity and acceleration • at time t and use them to analyze • the motion of the object.

  28. Example 2 – Solution • The velocity and acceleration are

  29. Example 2 – Solution • cont’d The object oscillates from the lowest point (s = 4 cm) to the highest point (s = –4 cm). The period of the oscillation is 2, which is the period of cos t.

  30. Example 2 – Solution • cont’d • The speed is | v | = 4 | sin t |, which is greatest when | sin t | = 1, that is, when cos t =0. • So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t =0, that is, at the high and low points. • The acceleration a = –4 cos t =0 when s = 0. It has greatest magnitude at the high and low points.

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