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Derivatives of Functions

Find and simplify derivatives of the functions: a) y = cos x ¯¯¯¯¯¯¯ x^2+e^2x b) y = ln((x^2 + 1)(cos x)) (^ means exponent and e^2x, 2x is the exponent)

Derivatives of Trigonometric Functions : Chain Rule

Use chain rule to verify that every function of the form y = a sin (5t) + b cos (5t) is a solution to the differential equation d^2y/dt^2 = -25y. Then use this fact to find the solution which also satisfies the initial conditions: y(0) = 3 and y'(0) = 0 (^ means exponent and d^2y is over dt^2)

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

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.


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


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)


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