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# Derivatives

### Derivatives: Minimizing Time and Distance

A lighthouse is 30 miles off a straight coast and a town is located 25 miles down the sea coast. Supplies are to be moved from the town to the lighthouse on a regular basis and at a minimum time. If the supplies can be moved at a rate of 4 miles per hour on water and 40 miles per hour on land, how far from the town should the do

### Derivative of a function where x is both base and exponent.

Find the derivative dy/dx if y = x^(x+3)

### Partial derivatives

The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation: &#8706;u/&#8706;t = (c^2)&#8706;^2u/&#8706;x^2 where t is the time, x distance from the beginning of the rod and c is the material constant. Function u(t,x) represents the temperature at the given time t and p

### Derivative using the product rule and the chain rule

Find the derivative of the function y=x^2e^(3x) See Attachment for a cleaner version of the question.

### Real Analysis Functions with Equal Derivatives

Let f be a function given by x + 2 if x < 0 f(x) = x if x >= 0 Is there a function g: R ---> R such that g'=f? *be careful applying definition of the derivative

### Differentiation : Existence of Solutions

I want to prove, for the numbers a and b, that the following equation has exactly three solutions if and only if 4a^3 + 27b^2 < 0: x^3 + ax + b = 0, x in R

### Derivatives and differentiation: Rate of Change

At what rate is the surface area of a cube changing the edge measures 5 inches and is changing at a rate of 2 in/min. GIVEN (A=6*s^2)

### Differentiation

Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial

### Partial Derivatives and Appropriate Chain Rule

Find partial deriv's w/r and w/theta using appropriate chain rule for : w=the square root of (25-5x^2-5y^2), where x=r cos theta, y= r sin theta.

### Find both first partial derivatives.

A) z=y^3-4xy^2-1 B) first partial derivative WRT x,y,z for : w=3xz/(x+y)

### Maclaurin Series Hyperbolic Tangent

Given The Maclaurin series for the inverse hyperbolic tangent is of the form x+x^3/3+x^5/5...x^7/7. Show that this is true through the third derivative term.

### Derivative of a Function: Product Rule

Use the product rule find derivative of h(t)= cubed root of t * (t^2+4)

### Partial Derivatives Functions

Find the first partial derivative with respect to x, y, z. w = 3xz / x+y

### Calculus : Partitions and Riemann Integral

Let f be the function: F(x) 1/4 x=o, x 0<x<1 3/4 x=1 Using standard partition Pn (0,1) where n greater or equal to 4 L(f, Pn) = 2n(squared) -3n+4 all divided by 4n(squared) U(f,Pn) = 2n(squared) +3n+4 all divided by 4n(squared) and deduce that f is intergrable on (0,1) and evaluate (intergral sign with 1 at t

### Working with derivatives and tangents

1.) Find the derivative of the function: a.) f(x) = x + 1/x^2 b.) f(x) = (2/3rd root of x) + 3 cos x 2.) Find equation of tangent line to the graph of f at the indicated point: a.) y = (x^2 + 2x)(x + 1) ; (1,6)

### Calculate the derivatives of wave equations

The equation for a wave moving along a straight wire is: (1) y= 0.5 sin (6 x - 4t) To look at the motion of the crest, let y = ym= 0.5 m, thus obtaining an equation with only two variables, namely x and t. a. For y= 0.5, solve for x to get (2) x(t) then take a (partial) derivative of x(t) to get the rate of change of

### Equating the function of a reversal tangent.

What is the equation of the reversal tangent of the following function: f(x)=-(x^3)+9x^2-(29x)+35

### Calculating the reversal tangents of a function.

What is the equation of the reversal tangent of the following function? f(x)=-(x^3)+9x^2-(29x)+35

### Implicit differentiation : the chain rule and product rule.

Use implicit differentiation to find dy/dx if y^2 + 3xy + x^2 + 10 = 0 (1) where y is a function of the independent variable x.

### First principles using standard theorems on trigonometric limits

Using the definition of the derivative and any standard limiting theorems, show that the derivative of (sinx)^2 is sin(2x).

### Derivatives: Chain Rule

Please see the attached file for the fully formatted problems. For a composite function f(x) = g(u(x)) state the chain rule for the derivative f(x). For each of the following functions, compute the derivative, simplifying your answers. f(x)=ln(1 + x^2) f(x)= sin(x^2) f(x) = (sin x)^2 (a) For a composite function f(x)

### Equation of a straight line which is tangent to a curve

For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1. See attachment for diagram.

### Using Cramer's Rule with a 3x3 System

Using Cramer's Rule with a 3x3 system 3x+4y+z=17 2x+3y+2z=15 x+y =4

### Working with inverse functions and their derivatives.

Given the function f(x) = e^(2x) a) Find the derivative b) Find the inverse (i.e. g(x)) c) Find the derivative of the inverse d) Find the value of g'(pi)

### Find a function whose derivative exists and is not continuous

Find a real valued function such that its derivative exists in every point, but it is not continuous at least in one point.

### Show me how to use the definition of the derivative.

The process of working with the definition of the derivative is shown using the example f(x)=3x-6.

### What am I doing when I take the derivative of a function?

The idea of the derivative is explained using the function x^2.

### Find the derivative of a quadratic polynomial.

Use the (limit) definition of the derivative to find the derivative of f(x)=3x^2-2x+1?

### Definition of derivative

Let f(x) be a continuous function of one variable. a) Give the definition of the derivative. b) Use this definition to find the derivative of f(x)=x^2+2x-5 c) Evaluate f'(2)

### First and second derivatives

Find the first and second derivatives of the function y=((1-x)/x^2)^3