Problem 6 Find $\frac{d y}{d x}$. $$y=... [FREE SOLUTION]

91影视

Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 1: Problem 6

Find $\frac{d y}{d x}$. $$y=7$$

Short Answer

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Step by step solution

- Understand the Problem

The problem requires finding the derivative $\frac{d y}{d x}$ of the given function $y = 7$.

- Recall the Derivative Rule for Constants

The derivative of a constant with respect to any variable is always zero. Mathematically, if $c$ is a constant, then $\frac{d c}{d x} = 0$.

- Apply the Derivative Rule

Since $y = 7$ is a constant, apply the rule: $\frac{d y}{d x} = \frac{d (7)}{d x} = 0$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

constant function

A constant function is a type of function where the output value does not change regardless of the input. In mathematical terms, if you have a function defined as $ y = c $, where $ c $ is a constant, the value of $ y $ will always be $ c $.

These functions are represented as horizontal lines on a graph. For example, consider the function $ y = 7 $. Here, no matter what value of $ x $ you plug in, the value of $ y $ remains 7.

Constant functions are simple to work with, especially when finding their derivatives.

derivative rules

Derivative rules are essential guidelines used to find the derivative of functions. Some basic rules make differentiation easier and more systematic.

Here are a few fundamental rules:

The Constant Rule: The derivative of a constant function, such as $ y = 7 $, is zero. Mathematically, if $ c $ is a constant, then $ \frac{d c}{d x} = 0 $.
The Power Rule: If you have a function $ y = x^n $, the derivative is given by $ \frac{d y}{d x} = n x^{(n-1)} $.
The Sum Rule: If you have a function that is the sum of two functions, such as $ y = f(x) + g(x) $, the derivative is the sum of the derivatives: $ \frac{d y}{d x} = f'(x) + g'(x) $.

Understanding these rules will help you master differentiation and solve various calculus problems effectively.

differentiation

Differentiation is the process of finding the derivative of a function. The derivative measures how a function changes as its input changes.

When you differentiate a function, you are essentially finding the rate at which the function's value is changing at any given point.

For example, let's differentiate the constant function $ y = 7 $. By applying the constant rule, we find that $ \frac{d y}{d x} = 0 $. This result tells us that the rate of change for $ y $ in relation to $ x $ is zero.

Differentiation is a crucial concept in calculus, helping to solve problems related to rates of change, velocities, and slopes.

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91影视

Find \(\frac{d y}{d x}\). $$y=7$$

Short Answer

Step by step solution

- Understand the Problem

- Recall the Derivative Rule for Constants

- Apply the Derivative Rule

Key Concepts

constant function

derivative rules

differentiation

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