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

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.

Derivatives

In Exercises 27–32, find dp/dq.

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

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 the derivatives of the functions in Exercises 19–40.

q = sin(t / (√t + 1))

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

___

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

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

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

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

𝔂 = x⁵ - 0.125x² + 0.25x

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

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 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)

Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

In Exercises 41–58, find dy/dt.

y = ((3t − 4) / (5t + 2))⁻⁵

Differentiating Implicitly

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

x²y + xy² = 6

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

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