Question

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ g(x)=-3 $$

Short answer

The derivative of the function \(g(x) = -3\) is \(g'(x) = 0\).

Step-by-step solution

Prepare the Equation

The derivative of a function \(f(x)\) can be defined by the limit process:\n\[f'(x)=\lim_{{h -> 0}} \frac{f(x + h) - f(x)}{h}\]\nSubstitute \(g(x) = -3\) into this formula to obtain the difference quotient.

Substitute the Function

Substitute \(g(x) = -3\) to the difference quotient:\n\[\lim_{{h -> 0}} \frac{g(x + h) - g(x)}{h} = \lim_{{h -> 0}} \frac{-3 - (-3)}{h}\]

Simplify the Difference Quotient

Simplifying above expression gives: \n\[\lim_{{h -> 0}} 0 = 0.\] The difference quotient simplifies to zero, because the numerator and denominator are the same, leading to the fraction being equal to zero.

The Final Derivative

The derivative \(g'(x)\) is 0 as the limit of the difference quotient for \(g(x)\) as \(h\) approaches 0 is 0.\n\[\ g'(x) = 0.\]

Key concepts

Difference Quotient

The difference quotient is a crucial concept in calculus that serves as the foundation for finding derivatives. It measures the rate at which a function's output changes with respect to a change in its input. To calculate the difference quotient, you need the function value at a certain point and its value at a point a small distance away, typically expressed as \( x+h \).

In the example of \( g(x)=-3 \), we use the difference quotient formula to set up the structure for finding the derivative: \[ \lim_{{h \rightarrow 0}} \frac{g(x + h) - g(x)}{h}.\] However, with a constant function like \( g(x)=-3 \), the output value does not change as 'x' changes, leading to the numerator of the difference quotient being zero whenever \( h \) is not zero. This simplicity is what ultimately allows us to determine the derivative of a constant function so directly.

Limit Definition of Derivative

Understandably, the limit definition of the derivative can seem abstract at first glance. It calculates the derivative of a function at a particular point based on the difference quotient and the limit as \( h \) approaches zero. The formal definition echoes the fundamental principle of calculus, where the derivative at \( x \) is the limit of the difference quotient: \[ f'(x) = \lim_{{h \rightarrow 0}} \frac{f(x + h) - f(x)}{h}. \]

In simple terms, this process measures how the function's rate of change at a specific point converges as the interval \( h \) becomes infinitesimally small. So when the function \( g(x) \) is constant, as in the illustrative exercise, we observe that the rate of change is nonexistent, hence, the derivative equals zero.

Simplifying Expressions

An integral part of working with derivatives is the ability to simplify expressions accurately and effectively. Simplifying can range from straightforward calculations to more complex algebraic manipulations. When applying the limit process to find a derivative, simplification often involves evaluating or canceling terms within the difference quotient.

In the context of the given exercise, simplifying involves noticing that the constant terms in the numerator cancel each other out: \( -3 - (-3) = 0 \). This results in the simplification of the difference quotient to zero, and thus the limit as \( h \) approaches zero also equals zero, since any term divided by \( h \) that results in zero signifies no change as \( h \) grows smaller.

Constant Function Derivative

A constant function is one where the output value does not depend on the input value; it remains the same regardless of the input. The derivative of a constant function, understandably, is always zero. This is because derivatives measure the change in a function's output relative to a change in the input, and with a constant function, there is no change to measure.

In the exercise where \( g(x) \) is defined as a constant function: \( g(x) = -3 \), the derivative \( g'(x) \) found through the limit process is zero. This reinforces the core concept that constant functions do not vary, and their slopes on a graph are always horizontal, corresponding to a derivative of zero.