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[DOC File]DERIVATIVES
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Definition: The derivative of a function f at a point a, denoted by f ′(a), is. provided that the limit exists. If we denote y = f (x), then f ′(a) is called the derivative of f, …
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[DOC File]AP Calculus Free-Response Questions
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Let g be a function whose derivative is given by g ’(x) = e-2x(3f(x) + 2f ’(x)) for all x. a. Write an equation of the line tangent to the graph of f at the point where x = 0. b. Is there sufficient information to determine whether or not the graph of f has a point of inflection . where x = 0? Explain your answer.
d dx e 2x

[DOCX File]UNIT 4.1 – Derivative of Exponential Functions
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The exact value for b is named "e" after the mathematician who discovered it, Leonhard Euler. e= When "e" is used as the base of an exponential function, we have a function that is its own derivative!
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[DOC File]Derivatives
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Let’s look at getting the derivative: The top polynomial is 2x + 6 and it’s derivative is 2. The bottom polynomial is x ( 3 and it’s derivative is 1. So, for example at x = 3 there is NO VALUE for the derivative…the graph is undefined at that place so there’s not a tangent to the not-graph. The graph is discontinuous there, too, remember?
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[DOC File]Math 121 - Calculus for Biology I - San Diego Miramar College
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Solution: The function f(x) can be considered a composite of the function f1(u) = u6 and the function f2(x) = x3 - 4x2 + e-2x. It is easy to find the derivatives of both f1 and f2. We have f1'(u) = 6u5 u' and f2'(x) = 3x2 - 8x - 2e-2x. From the chain rule, we see that . f '(x) = 6(f2(x))5f2'(x) = 6(x3 - 4x2 + e-2x)5(3x2 - 8x - 2e-2x).
e 2x graph

[DOC File]Practice Exercise Sheet 1 - TCD
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= (X 2 +3) (3X2) + (X 3 –1 ) (2X) + 12X = 3X 4 + 9X 2 + 2X 4 – 2X + 12X = 5X 4 + 9X 2 + 10X (vii) Y = ((X +1) (X 3 + 3X) can be re-written as Y = (X ½ +1) (X 3 + 3X) applying the product rule…. = (X ½ +1)( 3X 2 + 3) + (X 3 + 3X) (½ X –½ ) multiplying out…remember, xa.xb = x a+b so e.g. X ½. 3X 2 = 3X 5/2
how to find derivatives

[DOC File]Derivatives
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Ex. sin(xy) = 2x + 5. Note that y = sin u and u = xy, so , , so . Returning to the above implicit differentiation: Integration Methods. 1. Substitution. Let w be an “inside function” whose derivative, , will “cancel out” other terms. We then reassemble an integral of the form into the form . Ex. Let w = x3 , then and dw = 3x2dx. 2.
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[DOC File]Derivatives of Inverse Functions
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(e) Find , restricting it if necessary, and graph it. (f) Is a one-to-one function? ... Can you write a rule for finding the derivative of the inverse of a function without actually finding the inverse? _____ x. y. x. y. y. x. Title: Derivatives of Inverse Functions Author: 004ns485
second derivative of e 2x

[DOC File]AP Calculus Assignments: Derivative Techniques
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Review the two limit definitions of the derivative f '(a). 9. Kenny was given the function y = xlnx for x ( [1, e] and asked to find the rate of change of y with respect to x.
derivative of e power 2x

[DOC File]SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
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y(x) = c1 ex + c2 e3x + e–2x . is the solution of. y'' 4 y' + 3 y = 10 e–2x. where yh(x) = c1 ex + c2 e3x is the general solution of . y'' 4 y' + 3 y = 0. and yp(x) = e–2x satisfies the nonhomogeneous equation, i.e., yp(x) is a particular solution of the nonhomogeneous equation.
d dx e 2x

Derivative of e to the 2x