DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS

DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS

THE DERIVATIVE AS A FUNCTION f x = lim h 0 f x + h f(x) h Last class we examine the limit of the ifference quotient at a specific x as h 0, an relate it to the slope of the function at that value of x. If we generalize this result for all acceptable values of x we can fin an expression in x which represents the slope of the function at any x. Doing so gives us a function of x which we call a erivative. The process of obtaining the erivative of a function is calle ifferentiation.

EXAMPLES: For the following functions of x, ifferentiate to fin the erivative of the function at x = 0, x = 1, an x = 2: f x = x 2 + 2 What is the omain of the erivative of these functions? g x = 2 x 3 h x = x + 5

THE DERIVATIVE AS A FUNCTION Since the erivative is itself a function, f (x) will have: A omain & range A graph Intervals over which it is continuous Information about it s precursor f(x) Yes, a erivative of itself (calle the secon erivative more on this later)

DIFFERENTIABILITY OF A FUNCTION A function f(x) is ifferentiable at x = a if f (a) exists. It is ifferentiable on an open interval a, b if it ifferentiable at each number in the interval. Example: Suppose f x = 2 x2 x 2 +3x+2. What are the intervals over which f(x) is ifferentiable?

DIFFERENTIABILITY OF A FUNCTION Consier the following function of x: 2 x, x 1 f x = ቊ x 2, x > 1 Fin f 1 if possible, an fin a formula for f x. Is f(x) continuous at x = 1? Is it ifferentiable at x = 1? Theorem: If f(x) is ifferentiable at x = a, then f is continuous at x = a.

WHEN IS A FUNCTION NOT DIFFERENTIABLE? Whenever it is not continuous Whenever it has kinks, or sharp corners Consier the erivative of f x = x at x = 0 Whenever there are asymptotes Whenever there is a vertical tangent line (unefine slope) Whenever the function is too wiggly or oscillates too much Consier the erivative of f x = sin 1 x at x = 0 Whenever the function is bizarre nonsense Consier a function that is equal to 1 for rational numbers an 0 for irrational numbers

HIGHER ORDER DERIVATIVES If y = f x is ifferentiable, we can fin its erivative. We call this the secon erivative of f an write it as f x. In a future class, we will explore the sort of information f x can give us about f(x). Define as a limit, the secon erivative is the limit of the ifference quotient of ifference quotients. When simplifie, this is: f x = 2 f x + h 2f x + f(x h) f x = lim x2 h 0 h 2 (You will probably never nee this, but it s goo to know)

LAWS OF DIFFERENTIATION

DERIVATIVE OF A CONSTANT FUNCTION If f x = c where c is a constant, then f x = 0 This is because the slope of a horizontal line is always zero at every x, therefore the erivative of a horizontal line shoul always be zero, for all x.

POWER RULE If f x = x n where n is any real number, then f x = n x n 1 In other wors, the exponent in your original function becomes the coefficient in the erivative, an the exponent in the resulting erivative is reuce by 1. x x EXAMPLES: Fin the following erivatives. x x3 x 1 x x x x xπ x x 5 x 3

CONSTANT MULTIPLE RULE If c is a constant an f(x) is ifferentiable, then x cf x = c x f x In other wors, if our function of x is being multiplie by a constant, we can simply factor that constant out an multiply it to the erivative of the function. EXAMPLES: Fin the following erivatives. x 5x2 x 2 x x 10 x 2

SUM AND DIFFERENCE RULES If f(x) an g(x) are both ifferentiable, then x f x ± g x = x f x ± x g(x) In other wors, taking the erivative of the sum or ifference of two functions is the same as aing or subtracting the iniviual erivatives of each function. We can now use the combination of all these laws to fin the erivative of any polynomial function.

EXAMPLES: For f x = 2x 2 3x + 1, fin f x. For g x = 3 x 2x + 4, fin g 1. For h x = 4x 3 + 9x 2 12x + 3, calculate h x an etermine the values of x where f(x) has horizontal tangent lines (zero slope).

DERIVATIVES OF THE TRIG FUNCTIONS Proofs for these are long, but here are the erivatives for the trigonometric functions, sine an cosine: x sin x = cos x x cos x = sin x Note the negative!!

CLASSWORK & HOMEWORK MATH JOURNAL: Don t forget it!!! CLASSWORK: LAWS OF DIFFERENTIATION: For the polynomial function f x = x 3 8x 2 12x + 10, fin: a) The erivative f x b) The equation of the line tangent to f x at x = 1, in slope intercept form c) The values of x where f x has a horizontal tangent line Homework: Pg. 115, #2-54 (evens)