Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).

If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 2?

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.

f(x) = √(x + 1), (8, 3)

Find dr/dθ in Exercises 15–18.

r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

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.

(y - x)² = 2x + 4, (6, 2)

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

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

______

𝓻 = √2θ sinθ

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)

Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (−π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

The general polynomial of degree n has the form

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,

where aₙ ≠ 0. Find P'(x).

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.

2√y = x – y

In Exercises 41–58, find dy/dt.

y = (t⁻³/⁴ sin(t))⁴/³

Assume that y = 5x and dx/dt = 2. Find dy/dt

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

In Exercises 41–58, find dy/dt.

y = 4 sin(√(1 + √t))

Differentiating Implicitly

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

x cos(2x + 3y) = y sin x

Derivatives

In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)

Moving along a parabola A particle moves along the parabola y = x² in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m/sec. How fast is the angle of inclination θ of the line joining the particle to the origin changing when x = 3 m?

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) = x⁻¹, x₀ = 0.5, dx = 0.1

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

𝔂 = x³ - 3 (x² + π²)

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

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { x²/³, x ≥ 0

x¹/³, x < 0

In Exercises 43–50, find by implicit differentiation.

__

√xy = 1

Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.

Tangent Lines

In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.

y = sin x, −3π/2 ≤ x ≤ 2π

x = −π, 0, 3π/2

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 derivatives of the functions in Exercises 1–42.

__

𝔂 = ( √ x )²

( 1 + x )

The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = 3 csc(1 − 2√x)