The Differentiation Laws
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1 The Differentiation Laws Proofs September 21, The notation F x = L is read the it of F x as x approaches a is L. It means F x gets closer and closer to L as x gets closer and closer to a. Sometimes the textbook writes this as F x L as x a; this is read as F x approaches L as x approaches a or F x goes to L as x goes to a. I like to explain its by writing F x L when x a. The notation A B means A is approximately equal to B. 2 Here are two ways of writing the definition of the derivative: f fx + h fx x =, 1 h 0 h f a. 2 To use equation 1 to estimate f 4 we might take x = 4 and h = , so x + h = and f 4 = f x fx + h fx h = f4.001 f To use equation 2 to estimate f 4 we might take a = 4 and x = , so = and f 4 = f a 1 = f4.001 f
2 This illustrates that the two formulas are different ways of expressing the same thing. In equation 1 the variable h is a dummy variable whereas the variable x is a free variable. If all occurrences of dummy variable in an expression are changed to another letter, the meaning of the expression is unchanged: fx + k fx k 0 k fx + h fx =. h 0 h A free variable is used to assert that an equation is valid for a range of values; if the free variable is changed on one side of an equation it must be changed on the other side as well. Thus 1 could be written f fy + h fy y =. h 0 h One can substitute a number for a free variable as in f f4 + h f4 4 =, h 0 h but substituting a number for a dummy variable yields nonsense. In equation 2, x is the dummy variable and a is the free variable. 3 Definition. The derivative f a is defined by f a. D A function is differentiable at a iff this it exists and is finite. A function is differentiable iff it is differentiable at every a in its domain. A function f is continuous at a iff fx = fa. C A function is continuous iff it is continuous at every a in its domain. A function is continuous on a set e.g. interval I iff it is defined 1 and continuous at every point a of I. 1 i.e. I is a subset of the domain of f 2
3 4 The Limit Laws. These are c = c x = a F x + Gx = F x + Gx F x Gx = F x Gx F x Gx = F x Gx gf x = g F x Gx = F x Gx F x constant law identity law sum law difference law product law quotient law continuity law assuming that the its on the right exists. In the constant law c denotes a constant function, i.e. the function whose value is always the number c. In the quotient law we must also assume that the it in the denominator is nonzero. In the continuity law the function g is continuous. 5 Theorem. A differentiable function is continuous. Proof: Assume that the it D exists. Then = fx fa = const diff = hsa prod = 0 const, diff, ident = f a 0 definition = 0 hsa 3
4 so Equation C holds. In this proof const means constant law, diff means difference law, ident means identity law, prod means product law, and hsa means high school algebra. 6 The differentiation formulas are used to differentiate cu, u+v, u v, u v, and u/v when you know how to differentiate u and v. Where c is a constant, u and v are functions, and denotes differentiation the differentiation formulas are c = 0 constant formula u ± v = u ± v sum formula uv = u v + uv u v = u v uv v 2 product formula quotient formula The constant formula and the product formula imply cu = cu Using the product formula repeatedly gives d dx un n 1 du = nu dx. Using the quotient formula with numerator 1 proves that the formula works for negative integers. The quotient formula can be derived 2 from the product formula as follows: if w = u/v then wv = u so w v + wv = u so w = u wv v = u u/vv v = u v uv v 2. Using these formulas you can easily differentiate any rational function i.e. ratio of polynomials. 2 Unlike the argument below, this argument does not prove that w is differentiable if u and v are. 4
5 7 Proof of the constant formula. Suppose that fx = c for all x where c is a constant. Then f a = c c = 0 definition hypothesis hsa = 0 const 8 Derivative of identity function. Suppose that fx = x for all x. Then f a = = 1 definition hypothesis hsa = 1 const 5
6 9 Proof of the sum formula. Suppose that fx = ux + vx for all x where u and v are differentiable. Then f a = ux + vx ua + va ux ua vx va + ux ua vx va + definition hypothesis hsa it law = u a + v a definition 10 Proof of the product formula. Suppose that fx = uxvx for all x where u and v are differentiable. Then f a = ux vx ua va ux ua vx va va + ux ux ua vx va = va + ua definition hypothesis hsa it laws = u ava + v aua definition In the fourth step the theorem that a differentiable function is continuous is also used. 6
7 11 Proof of the quotient formula. Suppose that fx = ux/vx for all x where u and v are differentiable and va 0. Then f a = = = ux/vx ua/va uxva uavx vxva ux ua va vxva = ux ua = u a va va 2 ua va 2 v a = u ava uav a va 2 ua vxva va va 2 ua va 2 vx va vx va definition hypothesis hsa hsa law definition hsa In the fifth step the theorem that a differentiable function is continuous is also used. One of the following questions will appear on the first exam. 12 Question. Define differentiable function, continuous function, and prove that a differentiable function is continuous. In your proof you may use without proof the it laws and high school algebra. Answer: Write Equation D, Equation C, and the proof in Question. State and prove the formula for the derivative of a constant function. In your proof you may use without proof the it laws and high school algebra. Answer: Write what is in 7. 7
8 14 Question. State and prove the formula for the derivative of the function fx = x. In your proof you may use without proof the it laws and high school algebra. Answer: Write what is in Question. State and prove the formula for the derivative of the sum of two functions. In your proof you may use without proof the it laws and high school algebra. Answer: Write what is in Question. State and prove the formula for the derivative of the product of two functions. In your proof you may use without proof the it laws, the theorem that a differentiable function is continuous, and high school algebra. Answer: Write what is in Question. State and prove the formula for the derivative of the quotient of two functions. In your proof you may use without proof the it laws, the theorem that a differentiable function is continuous, and high school algebra. Answer: Write what is in 11. 8
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Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important