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Derivatives

Implicit differentiation

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.

Limit

Use L'Hopital's rule to evaluate the limit lim e^x - x - 1 x→0 ¯¯¯¯¯¯¯¯¯ x^2

Derivatives

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)

Find two 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)

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)

Derivatves

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¯¯¯

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:

4 questions on derivatives/limits

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

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,

Partial Derivatives

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, di

Vectors and Partial Derivatives

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

Vectors and Derivatives

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

Curve Sketching

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

Laws of Logarithms and derivative

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

Derivatives

Find the differential dy of the function y = &#12310;(1-3x^2)&#12311;^(-2). Please use derivative to solve & show steps/work.