### Derivatives (5 Problems)

Find the derivative of the function and simplify. f(x)= x^3(5 - 3x^2) f(x)= x^2 + x - 1/x^2 - 1 f(x)= (5x^2 + 2)^3 g(x)= x sqrt(x^2 + 1) given f(x)= 3x^2 + 7x +1, find f''(x)

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Find the derivative of the function and simplify. f(x)= x^3(5 - 3x^2) f(x)= x^2 + x - 1/x^2 - 1 f(x)= (5x^2 + 2)^3 g(x)= x sqrt(x^2 + 1) given f(x)= 3x^2 + 7x +1, find f''(x)

Use the definition of the limit to find the derivative of the function f(x) = 7x + 3 Find the slope of the graph of F at the indicated point f(x) = sqrt(x) + 2; (9,5) use the derivative to find the equation of the tangent line to the graph of f at the indicated point. f(x) = x^2 + 3/x; (1,4)

Find the derivative of the functions. y= (2x - 7)^3 h(x)= (6x - x^3)^2 f(t)= sqrt(t + 1) f(x)= x^3(x - 4)^2

Find the value of the derivative of the function at the indicated point. f(x)= 1/3 (2x^3 - 4) point (0, -4/3) h(x)= x/(x - 5) point (6,6) f(t)= (2t^2 - 3)/3t point (2, 5/6)

G(t)= 3t^2 / sqrt (t^2 + 2t - 1) The number N of bacteria in a culture after t days is modeled by: N= 400 [ 1- 3/ (t^2 + 2)^2] Complete the table. What can you conclude? t 0 1 2 3 4 dN/dt _ _ _ _ _

Find the derivative of the functions. f(x)= x(3x - 9)^3 y= x sqrt(2x + 3) y= t^2 sqrt (t - 2)

Y= 4x^3/2 /x

Find the derivative of the functions. y= 1/ x-2 s(t)= 1/ t^2+ 3t- 1 f(x)= 1/ (x^2- 3x)^2 y= 1/ (sqrt x + 2)

Find the derivatives of the functions y=x^2 + 2x/x y=3x^2 - 4x/6x f(x)= 3^sqrt x (sqrt x + 3) f(x)= 3x- 2/2x - 3

4.) Ever wonder why bacteria are so hard to contain ? A certain coloney of bacteria has an initinal population of 10,000. After t hours, the coloney has grown to a number P(t) given by P(t) = 10000(1+.86t + t^2). a.) Find the growth rate (rate of change) of the population P with respect to time t. [FInd P'(t)] b.) Fi

Please see the attached file for full problem description. 3. Use Cauchy-Riemann equations and the given theorem to show that the function _ f (z) = e^z is not analytic anywhere. Theorem: Suppose that f (z) = u (x, y) + i v (x, y) and that f'(z) exists at a point z0 = x0 + i y0. Then the first-

Show that the composite function G (z) = g (2z - 2 + i) is analytic in the half plane x > 1, with derivative .... see attachment

Consider a Lagrangian system, with configuration space R^n, given by (x^1, ... x^n); and Lagrangian L(x', ..., x^n; v^1, ... v^n). Now consider a new system of coordinates, (y^1,... ^n), for this same system, so the y's are functions of the x's; and, inverting, the x's are also functions of the y's. Find the Lagrangian in the y-

41) Suppose that the average yearly cost per item for producing x items of a business product is C(x)=10+(100/x) . if the current production is x=10 and production is increasing at a rate of 2 items per year, find the rate of change of the average cost. 45) Suppose a 6ft tall person is 12 ft away from a 18-ft tall lamppost. i

Find the Derivative y'(x) implicitly. 1) 3xy³-4x = 10yy² 2) sinxy= x² - 3 3) 3x+y³-4y = 10x² 4) xe^(power y) -3ysinx = 1 5) cos y - y² = 8 6) e^x² - 3y = x² + 1 Find the Equation of the Tangent Line at the Given Point. 7) x³ - 4y² = 4 at ( 2,1) 8) x³y² = -3xy at (-1

What is the derivative of f(x) = 4x(x2 + 1)3 ?

What is the derivative of f(x) = sin(xx) for x > 0?

Let f(x) = sin(2x + 1) and g(x) = x3 + 3 for all real x. Which of the following is equal to the derivative of the composite function f[g(x)]?

Let f(x) = 2sin x. Then f"(0) =

Let f(x) =x^2 ln x -(x^3 + 3)/2x for x>0. Then f'(x) =

If f(x) = xe^x/sin(x) for 0<x<PI, then f' (x) = ?

The volume (in gallons) of water in a tank after t hours is given by f(t) = 600 sin^2(Pi*t/12) for 0 <= t <= 6. What is the rate of flow of water into the tank, in gallons per hour?

Give a simple proof or counterexample to disprove: If tangent vector p E Rn is such that the directional derivative of f by k vanished for every function f then k =0. If function f on Rn is such that the directional derivative of f by every tangent vector at every point vanishes, then f is constant. The directional deri

If f(x) = x^4 - 4x^3 + 10 find the relative extrema of the function and the points of inflection of its graph. Also, sketch the graph of the function. a. The x-value(s) of the relative minima of function f: ________________ b. The x-value(s) of the relative maxima of function f: _______________ c. The x-value(s) of poi

12. Use implicit differentiation to find the slope of the curve y^2 - y + 2x = 0 at the point (x,y)=(0,1). (See attachment for full questions) 13. A spherical balloon is leaking...

Find the absolute maximum and minimum of f(x)= x^3 - 3x for -1<=x<=3. (See attachment for full questions)

X^2-2xy+y^3=5 find dy/dx using implicit differentiation using product rule

Please assist me with the attached problems, relating to first principles and Leibniz Notation.

Please See Attachment. Suppose f: R  R is differentiable and let Show that

Let (see equation in attached file) - find all the directional derivatives of f at 0 - is f continuous at 0? - Is f differentiable at 0?