f(x) f(a) Limit definition of the at a point in slope notation.

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1 Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p Definition. Limit Definition of Derivatives at a point for Single-Variable Functions Let f : D R be a single variable function, where D = Dom(f) R. Then, the erivative of a function f at a number a D, enote f (a) is given by f (a) = lim x a f(x) f(a) x a Limit efinition of the at a point in slope notation. = lim h 0 f(a + h) f(a) h Limit efinition of the at a point in erivative notation. The tangent line to the single variable function y = f(x) at point (a, f(a)) is the line through (a, f(a)) whose slope is equal to f (a), the erivative of f(x) at x = a. We call the erivative f (a) the instantaneous rate of change of y = f(x) with respect to x at x = a. Definition. Limit Definition of Derivatives as a function for Single-Variable Functions Let f : D R be a single variable function, where D = Dom(f) R. Then, the erivative of a function f at any number x D, enote f (x) is given by f (x) = lim h 0 f(x + h) f(x) h Limit efinition of the erivative in erivative notation. The function f (x) is a function of variable x, known as the erivative of f with respect to x.

2 Theorem. Differentiability Implies Continuity If f is ifferentiable at a, then f is continuous at a. Theorem. Increasing Test an Decreasing Test Increasing Test: If f (x) > 0 on interval I D, then f is increasing on this interval. Decreasing Test: If f (x) < 0 on interval I D, then f is ecreasing on this interval. Theorem. Concave Up Test an Concave Down Test Concave Up Test: If f (x) > 0 on interval I D, then f is concave up on this interval. Concave Down Test: If f (x) < 0 on interval I D, then f is concave own on this interval. Theorem. Derivative Rules for Algebraic Combinations of Functions Suppose that a are real numbers an f an g are ifferentiable functions. Further, for the chain rule suppose that f is ifferentiable at g(x). Then, as long as we check these conitions, we can conclue 1. Constant Multiple Rule: x [c f(x)] = c x [f(x)] 2. Sum/Difference Rule: [f(x) ± g(x)] = x x [f(x)] ± x [g(x)] 3. Prouct Rule: x [f(x) g(x)] = f(x) x [g(x)] + g(x) x [f(x)] 4. Quotient Rule: x [ ] f(x) = g(x) g(x) x [f(x)] f(x) x [g(x)] [g(x)] 2 5. Chain Rule: x [f(g(x))] = f (g(x)) f (x)

3 Theorem. Derivative Rules for Power, Exponential an Logarithmic Functions Suppose that a, c are real numbers an n R. Then, as long as we check these conitions, we can conclue Constant Rule: Simple Power Rule: General Power Rule: Derivative of Exponent Rule: Derivative of Natural Exponent Rule: Derivative of ln( x ): Derivative of log a (x): x [c] = 0 If n N, then x [xn ] = n x n 1 If n R, then x [xn ] = n x n 1 x [ax ] = a x ln(a) x [ex ] = e x x [ ( ) ] ln x = 1 x [ ] log x a (x) = 1 x ln(a)

4 Theorem. Derivative Rules for Trigonometric Functions Suppose that a, c are real numbers an n R. Then, as long as we check these conitions, we can conclue Derivative of sin(x) Rule: Derivative of cos(x) Rule: Derivative of tan(x) Rule: Derivative of csc(x) Rule: Derivative of sec(x) Rule: Derivative of cot(x) Rule: [sin(x)] = cos(x) x [cos(x)] = sin(x) x x [tan(x)] = sec2 (x) [csc(x)] = csc(x) cot(x) x [sec(x)] = sec(x) tan(x) x x [cot(x)] = csc2 (x) Theorem. Derivative Rules for Trigonometric Functions Suppose that a, c are real numbers an n R. Then, as long as we check these conitions, we can conclue Derivative of arcsin(x) Rule: Derivative of arccos(x) Rule: Derivative of arctan(x) Rule: Derivative of arccsc(x) Rule: Derivative of arcsec(x) Rule: Derivative of arccot(x) Rule: x [arcsin(x)] = 1 1 x 2 x [arccos(x)] = 1 1 x 2 x [arctan(x)] = x 2 x [sec(x)] = 1 x x 2 1 x [sec(x)] = 1 x x 2 1 x [cot(x)] = x 2

5 Proceure. Implicit Differentiation Implicit ifferentiation is a technique use to fin y = y when we are consiering an equation x involving variables x an y such that y cannot be written explicitly in terms of x. In this case, we apply implicit ifferentiation using a number of steps: 1. Take erivative of both sies of implicit equation with respect to x 2. Assume that output variable y is an explicit function of input variable: x: i.e. assume y = y(x) 3. Apply chain rule an other rules of ifferentiation to both sies of equation. For the erivative of the inner function, introuce algebraic symbol representing unknown erivative: y = y x. 4. Solve for unknown an esire erivative. Proceure. Logarithmic Differentiation Logarithmic ifferentiation is a technique use to fin simplify our process of taking erivatives of complex algebraic function by relying on rules of logarithms: 1. Take natural log of both sies of equation y = f(x) 2. Use rules of logarithms to simplify expressions on both the left an right han sie of equality. 3. Differentiate implicitly both sies of equations with respect to x 4. Solve for unknown an esire erivative.