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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.3.9
Chapter 3, Problem 3.3.9

Derivative Calculations


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


y = 6x² − 10x − 5x⁻²

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1
Step 1: Identify the function y = 6x² − 10x − 5x⁻². We need to find the first derivative, which involves differentiating each term separately.
Step 2: Differentiate the first term 6x². Using the power rule, the derivative of x^n is n*x^(n-1). Therefore, the derivative of 6x² is 12x.
Step 3: Differentiate the second term -10x. The derivative of x is 1, so the derivative of -10x is -10.
Step 4: Differentiate the third term -5x⁻². Again, using the power rule, the derivative of x⁻² is -2*x⁻³. Therefore, the derivative of -5x⁻² is 10x⁻³.
Step 5: Combine the derivatives from steps 2, 3, and 4 to find the first derivative: y' = 12x - 10 + 10x⁻³. To find the second derivative, differentiate y' using the same rules applied to each term.

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Key Concepts

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

Derivative

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The first derivative indicates how the function is changing, while the second derivative provides insight into the curvature or concavity of the function.
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Power Rule

The Power Rule is a basic differentiation technique used to find the derivative of functions in the form of ax^n, where a is a constant and n is a real number. According to this rule, the derivative is calculated by multiplying the coefficient by the exponent and then reducing the exponent by one. This rule simplifies the process of finding derivatives for polynomial functions, making it essential for solving problems involving such expressions.
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Concavity and Inflection Points

Concavity refers to the direction in which a function curves, determined by the sign of the second derivative. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down. Inflection points occur where the concavity changes, which can be identified by setting the second derivative equal to zero. Understanding concavity is crucial for analyzing the behavior of functions and their graphs.
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Related Practice
Textbook Question

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

_____

𝔂 = / x² + x

√ x²

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Textbook Question

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.

Textbook Question

In Exercises 19–22, find the slope of the curve at the point indicated.


y = (x − 1) / (x + 1), x = 0

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Textbook Question

Estimating height of a building A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%?

Textbook Question

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.

Textbook Question

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.


y² – 2x = 1 – 2y