3. Techniques of Differentiation

Find the derivatives of the functions in Exercises 1–42.

𝔂 = (x + 1)² (x² + 2x)

Find the derivatives of the functions in Exercises 1–42.

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s = √ t .

1 + √ t

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

b. ƒ(x)g²(x), x = 0

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

c. ƒ(x) , x = 1

g(x) + 1

Find the derivatives of the functions in Exercises 1–42.

𝔂 = 1 x² csc 2

2 x

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x⁻¹/² sec (2x)²

Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).

Generalizing the Product Rule The Derivative Product Rule gives the formula

d/dx (uv) = u (dv/dx) + (du/dx) v

for the derivative of the product uv of two differentiable functions of x.

b. What is the formula for the derivative of the product u₁u₂u₃u₄ of four differentiable functions of x?

Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,

d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹

where m is a positive integer.

Assume that functions f and g are differentiable with f(1) = 2, f'(1) = −3, g(1) = 4, and g'(1) = −2. Find the equation of the line tangent to the graph of F(x) = f(x)g(x) at x = 1.

Assume that functions f and g are differentiable with f(2) = 3, f'(2) = −1, g(2) = −4, and g'(2) = 1. Find an equation of the line perpendicular to the line tangent to the graph of F(x) = (f(x) + 3) / (x − g(x)) at x = 2.

Find the derivatives of the functions in Exercises 17–28.

y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

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Find the derivatives with respect to x of the following combinations at the given value of x.

c. f(x) / (g(x) + 1), x = 1