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

s = cos⁴ (1 - 2t)

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

𝔂 = 1 x² csc 2

2 x

The diameter of a sphere is measured as 100 ± 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = √u, u = sin x

Differentiating Implicitly

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

x⁴ + sin y = x³y²

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.

Finding Linearizations

In Exercises 1–5, find the linearization L(x) of f(x) at x = a.

f(x) = ∛x, a = −8

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

x²/³ + y²/³ = 1

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

lim (x → −1) (x²/⁹ − 1) / (x + 1)

A balloon and a bicycle A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance s(t) between the bicycle and the balloon increasing 3 sec later?

Theory and Examples

Intersecting normal line The line that is normal to the curve x² + 2xy – 3y² = 0 at (1,1) intersects the curve at what other point?

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

_____

𝔂 = / x² + x

√ x²

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

y = 6x² − 10x − 5x⁻²

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

3 + sin y = y – x³

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

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Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = cot(πu/10), u = g(x) = 5√x, x = 1

In Exercises 41–58, find dy/dt.

y = (t tan(t))¹⁰

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

Find the first and second derivatives of the functions in Exercises 33–38.

s = (t² + 5t − 1) / t²

Is there a value of b that will make

g(x) = { x + b, x < 0

cos x, x ≥ 0

continuous at x = 0? Differentiable at x = 0? Give reasons for your answers.

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 surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

r = 12/θ − 4/θ³ + 1/θ⁴

In Exercises 53 and 54, find dr/ds.

2rs - r - s + s² = -3

Normal lines parallel to a line Find the normal lines to the curve xy + 2x – y = 0 that are parallel to the line 2x + y = 0.

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

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))