Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.31

Chapter 3, Problem 3.31

Find the derivatives of the functions in Exercises 1–42.
__
𝔂 = ( √ x )²
( 1 + x )

Verified step by step guidance

First, simplify the expression \( y = (\sqrt{x})^2(1 + x) \). Note that \((\sqrt{x})^2 = x\), so the function simplifies to \( y = x(1 + x) \).

Next, expand the expression \( y = x(1 + x) \) to make differentiation easier. This gives \( y = x + x^2 \).

Now, differentiate the function \( y = x + x^2 \) with respect to \( x \). Use the power rule for differentiation, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \).

Apply the power rule to each term: The derivative of \( x \) is \( 1 \), and the derivative of \( x^2 \) is \( 2x \).

Combine the derivatives of each term to find the derivative of the entire function: \( \frac{dy}{dx} = 1 + 2x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are denoted as f'(x) or dy/dx.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(x), the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is essential when differentiating functions that are nested within each other.

Power Rule

The power rule is a basic rule for finding the derivative of a function in the form of f(x) = x^n, where n is a real number. According to this rule, the derivative is given by f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions and is widely used in calculus.

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

Textbook Question

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?

Textbook Question

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

Textbook Question

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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Textbook Question

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.

x³/² + 2y³/² = 17, (1, 4)

Textbook Question

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

𝔂 = (θ² + sec θ + 1)³

Find the derivatives of the functions in Exercises 1–42.__𝔂 = ( √ x )²( 1 + x )

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Key Concepts

Derivative

Chain Rule

Power Rule

Find the derivatives of the functions in Exercises 1–42.
__
𝔂 = ( √ x )²
( 1 + x )