Find all points on the curve y = tan x, −π/2 < x < π/2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent lines together, labeling each with its equation.

Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

g(x) = 1 + √x

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

___

𝓻 = sin √ 2θ

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.

y = (2x + 1)⁵

Slopes and Tangent Lines

In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

g(x) = 8 / x², (2, 2)

In Exercises 41–58, find dy/dt.

y = (1 + tan⁴(t/12))³

Derivatives

In Exercises 27–32, find dp/dq.

p = (sin q + cos q) / cos q

Find the derivatives of the functions in Exercises 19–40.

y = (4x + 3)⁴(x + 1)⁻³

Find dr/dθ in Exercises 15–18.

r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

For what value of c is the curve y = c/ (x + 1) tangent to the line through the points (0, 3) and (5, -2)?

Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the volume V = (4/3)πr³ of a sphere when the radius changes from r₀ to r₀ + dr

Slopes, Tangent Lines, and Normal Lines

In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.

y = 2 sin(πx – y), (1,0)

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x²(x – y)² = x² – y²

Finding g on a small airless planet Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m/sec. Because the acceleration of gravity at the planet’s surface was gₛ m/sec², the explorers expected the ball bearing to reach a height of s = 15t − (1/2)gₛt² m t sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of gₛ?

Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ

[Technology Exercise]

Graph the curves in Exercises 39–48.

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.

y = 4x²/⁵ − 2x

For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.

lim (x → 1) (x⁵⁰ − 1) / (x − 1)

a. Graph the function

ƒ(x) = { x², -1 ≤ x < 0

{ -x², 0 ≤ x ≤ 1.

b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?

Give reasons for your answers.

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

Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

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

__

𝔂 = ( √ x )²

( 1 + x )

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = (1 / x³), (−2, −1/8)

Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = (nRT / (V − nb)) − (an² / V²),

in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)

<IMAGE>

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

__

x + √xy = 6, (4, 1)

In Exercises 19–22, find the slope of the curve at the point indicated.

y = x³ − 2x + 7, x = −2

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

f(x) = x⁴, x₀ = 1, dx = 0.1

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x²y + xy² = 6

Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.