Finding Derivative Functions and Values 

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s

b. volume?

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

Theory and Examples

The equations in Exercises 49 and 50 give the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.

s = 2 − 2 sin t

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)

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

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder’s volume is V = πh³. The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h.

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

q = sin(t / (√t + 1))

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

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𝓻 = sin √ 2θ

If r = sin(f(t)), f(0) = π/3, and f'(0) = 4, then what is dr/dt at t = 0?

Derivatives

In Exercises 1–18, find dy/dx.

f(x) = x³ sin x cos x

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?

Graphs

Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d).

Find dr/dθ in Exercises 15–18.

r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

Derivatives

In Exercises 27–32, find dp/dq.

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

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)

Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ

Differentiating Implicitly

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

x²y + xy² = 6

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

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

Differentiating Implicitly

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

x³ + y³ = 18xy

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)

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

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 = ((3t − 4) / (5t + 2))⁻⁵

Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

In Exercises 41–58, find dy/dt.

y = (1 + cos(2t))⁻⁴

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

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

In Exercises 41–58, find dy/dt.

y = (1 + tan⁴(t/12))³