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Write down the derivative of each of the following functions. f(x)=e^-2x. (thats e to the power minus2x). g(x)=sin(7x). Hence by using the product rule,differtiate k(x)=e^-2xsin(7x)

Determine derivative

(See attached file for full problem description) Determine the derivative: 1) d/dx

Differentiation of composite function - integral form

(See attached file for full problem description with proper symbols) --- Assume that f is continuous on [a,b], g is differentiable on [c,d], g([c,d]) [a,b] and F(x) = For each x [c,d]. Prove that F'(x)=f(g(x))g'(x) For each x (c,d).


Differentiate: 1) y=2x^2+5x+1 at x=5 2) y=1 - x - x^3 at x=-3 3) y = (1/x^2) - 2x^3 at x = -1

Revenue, Supply and Demand Functions : Derivatives and Integrals

5. A man was sentenced to 50 years in prison when he was 20 years old. While in prison he reflected on his life and decided that he should turn his life around and do something good for his society. He then became a model prisoner and his good behavior earned him the privilege to pursue a career in law. When he became 39 years

Implicit differentiation

Consider this equation: x2 - 2xy + 4y2 = 64 A) write an expression for the slope of the curve at any point (x,y) B) Find the equation of the tangent lines to the curve at the point x = 2 C) find d2y/dx2 at (0,4)

Critical Numbers, Derivatives and Rates of Change

The function has one critical number. Find it. A student decided to depart from Earth after his graduation to find work on Mars. Before building a shuttle, he conducted careful calculations. A model for the velocity of the shuttle, from liftoff at t = 0 s until the solid rocket boosters were jettisoned at t = 60.7 s, is gi

Applications of Derivatives Word Problems and Rate of Change

A street light is at the top of a 14 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 40 ft from the base of the pole? Note: You should draw a picture of a right triangle with the vertical side representing the pole,

Interest and applications of derivatives.

A person's fortune increases at a rate to the square of they're present wealth. If the person had one million dollars a year ago and has two million today then how much will the person be worth in six months?

Implicit Differentiation

Use implicit differentiation to find the slope of the tangent line to the curve at the point . Find by implicit differentiation. Match the expressions defining implicitly with the letters labeling the expressions for . 1. 2. 3. 4. A. B. C. D. Let Let Let Then

Directional Derivatives

Consider the function f(x,y,z) = (e^z)ln(x^2 + y^2) a) Is there a vector r such that the directional derivative of f at (1,1,0) in the direction of r equals 1? If there is, find one such vector. If not, explain why not. b) Is there a vector r such the directional derivative of f at (1,1,0) in the direction of r equals to

Partial Differentiation and the Chain rule

The problem states: Find dw/dt (a) using the appropriate chain rule and (b) by converting w to a function of t before differentiating. w = xy x = s sin t, y = cos t the solution in my solution manual goes like this: a) using the chain rule they come up with: 2y cos t + x(-sin t) = 2y cos t - x sin t = 2

Derivatives and Inverse Functions

Suppose g is the inverse function of a differentiable function f and let G(x) = 1/g(x), if f(3) = 2 and f'(3) = 1/9, find G'(2). Please see the attached file for the fully formatted problems.

Functions : Linear Regression, Derivatives and Rate of Change

1. A college calculus professor wanted to investigate the relationship between student's scores on the first exam and the overall course grades. A sample of the data is below. (All values are given in percents.) first exam score 54 98 73 100 88 90 77 73 81 final grade % 60 93 69 95 82 87 72 71 74