### Derivatives of Functions: Calculations

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

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

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

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)

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

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)

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)−1 w→0 ¯¯¯w¯¯¯ and this was the response: lim h(w)−1 w→0 ¯¯¯w¯¯¯

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

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

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:

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*

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

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

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

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. 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

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?

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,

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.

^ 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> .

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(

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.....

^ ^ ^ ^ 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

Show that z = x^(2) - y^(2) + 2xy satisfies Laplace's Equation z (subscript)xx + z (subscript)yy.

Verify: If w = tan (x^(2) + y^(2)) + x(y)^(1/2) , then w (subscript)xy = w (subscript)yx . keywords: differentiating, differentiate, differential

^ ^ ^ Find k(t) for r(t) = t^(2) i + t^(3) j . keywords: differentiating, differentiate, differential

^ ^ ^ Given r = 3 cos t i = 3 sin t j , find the velocity, acceleration and speed for an arbitrary time t.

^ ^ Find r '(t) for r (t) = < ( 4 + 5t), (t - t^2) >.

See attached file for full problem description. 1.Boat A is sailing south at 24 km/h while boat B, which is 48 kilometres due south of A, is sailing east at 18km/h. a)At what rate are they approaching or separating one hour later? b)When do they cease to approach on another and how far apart are they at this time? c)What

Please see the attached file for full description 1.Evaluate the following logarithms. a)log22log55 b)log23200-2log210 2.Solve the following logarithmic equation. log25=log2(x+32)-log2x 3.Determine the derivative of each of the following functions. a)y=ln(x2+3x+4) b)y=logx2 c)y=lnx^2/(3x-4)^3 d)y=2x3e4x

Find the differential dy of the function y = 〖(1-3x^2)〗^(-2). Please use derivative to solve & show steps/work.