### Derivatives and differentiation: Rate of Change - Surface Area of a Cube

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)

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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)

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

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.

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

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.

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

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

True or False and Why? 1.) If f(x) = g(x) + c, then f'(x) = g'(x) 2.) If y = x/pi, then dx/dy = 1/pi 3.) If f(x) = 1/x^n, then f'(x) = 1/(nx^n-1)

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)

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

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

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.

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

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)

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

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

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

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

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

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)

Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. Question: Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b]

Find the rule for the sequence below. square 2x2 =10 3x3=40 +30=20 difference 4x4=90+50=20 difference 5x5=160+70=20 difference

Find a polynomial p so that: p''(t)+3p'(t) + 2p(t) = (t^2)-2 for all numbers t. (note: p''= p double prime and t^2 = t raised to the power of 2)

Two carts A and B are connected by a rope 39 feet long that passes over pulley P. The point Q is on the floor directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q? Express solution using related rate n

Find the derivative. a) f= 4-sqrt(x+3) b) f= (x+1)/(2-x) See attachment below for additional information.