Explore BrainMass
Share

# Derivatives

### Functions : Derivatives, Areas of Increase and Extrema

Suppose f(x) = ln(2 + cos x) on the interval (0, 2pi). a) Calculate f' (x) and f" (x). b) Find the interval(s) on which the function f is increasing. c) Find all extreme values of f and the values of x at which they occur. d) Find the interval(s) on which the function f is concave up?

### Functions : Derivatives, Areas of Increase and Extrema

Suppose f(x) = ln(2 + cos x) on the interval (0, 2pi). a) Calculate f' (x) and f" (x). b) Find the interval(s) on which the function f is increasing. c) Find all extreme values of f and the values of x at which they occur. d) Find the interval(s) on which the function f is concave up?

### implicit differentiation and explicit function

Please explain the steps and solution, thanks: The equation 4x^2y - 3y = x^3 implicitly defines y as a function of x. a) use implicit differentiation to find dy/dx. b) write y as an explicit function of x and compute dy/dx directly.

### L'hopital's rule problem

Use L'Hopital's rule to evaluate the limit lim e^x - x - 1 x&#8594;0 ¯¯¯¯¯¯¯¯¯ x^2

### Derivatives Using Product and Chain Rules

Find and simplify the derivatives of the functions: a) k(x) = sin^7(2x) b) g(x) = xe^2x (^ means exponent and ^2x is the total exponent of xe)

### Calculate derivatives of the function

Calculate derivatives of the function f (x) = 2^x + x^2 + pi^2 + ln2

### Chain rule to find derivatives

Please explain the steps and solution: Calculate derivatives of the function h (x) = ln (x^3) - ln (^2) (^ means exponent)

### Finding derivative using quotient rule

Calculate the derivatives of the function: g (x) = x^2 - 5x/ &#8730;x (x^2 - 5x is over &#8730;x as in a fraction and ^ means exponent)

### Derivatives of Functions: Calculations

Calculate the derivatives of the functions: a) k (x) = 3^4&#8730;x^5 - 5/^3&#8730;x^2 (this means it is a fraction: 5 is over ^3&#8730;x^2)

### Find f' (x)

Please explain the steps and the solution, thank you: Find f' (x) of f(x) = sin (e^3x) (^ means exponent)

### Derivatives: Finding Functions

Please explain the steps and solution, thank you: Find two functions f and g such that the derivatives f' (x) = g' (x) = 3x^2 and such that f (x) - g (x) = 7 (^ means exponent)

### Find f'(x)

Please explain the solution, thank you: Find f'(x) of f(x) = sin (e^3x) (^ means exponent)

### Algebra: Derivatives and Tangent Lines

Suppose that derivative of the function f is given by f'(x) = 3x and suppose that the point (4,2) is on the graph of f. Write an equation of a line tangent to the graph of f at the point (4,2)

### Investigating Derivatives

Hi, this was the question h is a function such that h(0) = 1, h(2) = 7, h(4) = 5, h'(0) = -2, h'(2) = 3, and h'(4) = -1 Evaluate lim h(w)&#8722;1 w&#8594;0 ¯¯¯w¯¯¯ and this was the response: lim h(w)&#8722;1 w&#8594;0 ¯¯¯w¯¯¯

### Second Derivatives Problem

Let n(x) = 1/x. Find a formula for n''(x)

### Limit Definition of the Derivative

Let g(x) = x^2 + x. Use the limit definition of the derivative to show that g'(x) = 2x + 1

### Limits and Derivatives

Differentiate: Y=x^2+4x(x^3/2) NOTE: x^2 =(squared) x^3/2 =(x raised to the 3/2) Determine whether the limit exist, If so compute the limit: Lim (square root of x) minus 4 divided by x^3+18 x-->1 NOTE: x^3 = x cubed Find the Limit: Lim x--> 0 (1 divided by x^3 minus 1) +1 NOTE:

### Equation of the Line Tangent to the Curve

Hi, this was your response since dy/dx means the slope of a tangent to the curve at a certain point there fore dy/dx= (3 x^2)- (6*2*x^1) dy/dx = (3x^2) - (12x) Equation of the line tangent to the curve= dy/dx = 3x^2 - 12x Thanks Ramesh but can you please explain further how you got dy/dx= (3 x^2)- (6*

### Finding the Derivatives in Calculus

Q#8. For f(x)=2-5X^2, Find: i) f`(x). Answers: a. -5x b. -10x c. -10x d. 0 Q#9. Consider f(x)=3-4*Square root(X) i) Find f`(x) Answers: a. - square root of X / 2 b. - 4 / square root of X c. -2/square root of x d. -8 / Square root of x. Q#10. Consider f(x)=3x/x+9 i) find f`(x) Q#7. A per

### Determine the equation for the derivative dy/dx

Given the equation y = x , determine the equation for the derivative dy/dx, using x-5 the delta process or the definition. Determine the derivative y = ( 3x+4)^ (2/3) * (x-1), find dy/dx Given the implicit relation x^3 y^2 + 2y^3 = 4x^3 + 7, determine dy/dx A particle moves in metres according to the parameterized

### Derivatives : Velocity and Displacement

27. A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10 m/s. It strikes the ground with a speed of 60 m/s. How tall is the building? Answer: y0 = 178.57 m 35. A stone is dropped from rest at an initial height h above the surface of the Earth. Show that the speed wit

### Proofs: Polar Coordinates

Please view the attached file for proper formatting on the following questions regarding polar coordinates. 1. Consider the polar coordinates: x = rcos(theta) y = rsin(theta) Questions a) to e) can be seen in the attachment.

### Derivation using the product rule

Derivation using the product rule. See attached file for full problem description. Differentiation using the product rule: If we have then the differentiation will be in the form of: Differentiate: 1. and then we have 2. and then 3. This function is composed of summation of two functions wh

### Derivatives and Rate of Change

Early one morning it began to snow at a constant rate. At 7 AM a snowplow set off to clear a road. By 8 AM it had traveled 2 miles but it took two more hours for the snowplow to go another 2 miles. Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour), at what time did it start to snow?

### Derivatives, Rate of Change and the Fundamental Theorem of Calculus

1. The resale value of a certain industrial machine decreases at a rate that depends on its age. When the machine is t years old, the rate at which its value is changing is -960e^-t/5 dollars per year. a) Express the value of the machine in terms of its age and initial value. b) If the machine was originally worth \$5,200,

### Critical Points and Continuous Second Derivatives

2. Suppose (0, 2) is a critical point of a function g with continuous second derivatives. In each case, what can you say about g? (a) gxx(0, 2) = ? 1, gxy(0. 2) = 6, gyy(0, 2) = 1 Please see the attached file for the fully formatted problems.

### Calculus - Directional Derivative

^ Find the directional derivative of f at P in the direction of a , where f(x,y) = x^2 - 3xy + ^ 4y^3 ; P (-2,0) ; a = <1,2> .

### Partial Derivatives of Surface Equations and Clairaut's Theorem

Find all the second partial derivatives. 47. f(x, y) = x^4 - 3(x^2)(y^3) Verify that the conclusion of Clairaut's Theorem holds, that is, u_xy = u_yx. 55. u = ln[sqrt(x^2 + y^2)] Find the indicated partial derivative. 59. f(x, y, z) = cos(4x + 3y + 2z); f_xyz, f_yzz 89. If f(x,y) = x(x^2 + y^2)^-3/2 * e^(sin(

### Partial Derivatives Investigation

Verify: If f(x, y, z) = 2z^(3) - 3 (x^(2) + y^(2)) z , then F_xx + F_yy + F_zz = 0 . P.S. F_xx = F(subscript)xx, etc.....

### Derivatives of Vector Cross Products

^ ^ ^ ^ Calculate d/dt [ r_1(t) * r_2(t) ] for r_1(t) = < 2t, 3t^(2), t^(3) > , r_2(t) = < 0, 0, t^(4)> . P.S. * = cross product and r_1 = r(subscript)1, r_2 = r(subscript)2 keywords: differentiating, differentiate, differential