Questions from General Calculus

Q: (a). Use the Product Rule twice to prove that if

(a). Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh + fg'h + fgh'. (b). Taking f = g = h in part (a), show that d/ dx [f (x)]3 = 3[f (x)]2 f'(x) (c). Us...

Q: (a). If F (x) = f (x

(a). If F (x) = f (x) g (x), where f and g have derivatives of all orders, show that F" = f"g +2f'g' + fg". (b). Find similar formulas for f"' and F(4). (c). Guess a formula for F(R).

Q: Use the definition of a derivative to show that if f (

Use the definition of a derivative to show that if f (x) = 1/x, then f'(x) = -1/x2. (This proves the Power Rule for the case n = -1.)

Q: Differentiate. g (θ) = eθ (tan θ

Differentiate. g (θ) = eθ (tan θ – θ)

Q: (a). If g is differentiable, the Reciprocal Rule says

(a). If g is differentiable, the Reciprocal Rule says that Use the Quotient Rule to prove the Reciprocal Rule. (b). Use the Reciprocal Rule to differentiate the function in Exercise 16. Exercise 1...

Q: Find a second-degree polynomial P such that P (2

Find a second-degree polynomial P such that P (2) = 5, P'(2) = 3, and P"(2) = 2.

Q: The equation y" + y' -2y = x2 is called

The equation y" + y' -2y = x2 is called a differential equation because it involves an unknown function and its derivatives y' and y". Find constants A, B, and C such that the function y = Ax2 + BX +...

Q: Differentiate the function. f (x) = x2 -

Differentiate the function. f (x) = x2 -3x + 1/x2

Q: Use the result of Exercise 63(c) to find an

Use the result of Exercise 63(c) to find an antiderivative of each function. (a). f (x) = √x (b). f (x) = ex + 8x3 Exercise 63(c): (c). Find an antiderivative for f (x) xn, where n ≠ -1. Check by d...

Q: Differentiate the function. y = x2 + 4x +3

Differentiate the function. y = x2 + 4x +3/√x