Question

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=7 x-3 $$

Short answer

The derivative of the function \(f(x) = 7x - 3\) is 7.

Step-by-step solution

Function Definition

First, we need to define function properly, so our function is \(f(x) = 7x - 3\).

Difference Quotient Definition

Next we need to form the difference quotient according to the formula \(f'(x) = \lim_{h->0} \frac{f(x+h)-f(x)}{h}\).

Substitute the function into the Difference Quotient

We substitute the given function into this difference quotient formula, which gives \(f'(x) = \lim_{h->0} \frac{(7(x+h) - 3) - (7x - 3)}{h}\).

Simplify the Expression

After simplifying the expression in the numerator and denominator, we get \(f'(x) = \lim_{h->0} \frac{7h}{h}\). When we divide 7h by h, we get \(f'(x) = \lim_{h->0} 7.\)

Find the Limit as h approaches 0

Finally, we find the limit as \(h\) approaches 0, which gives us \(f'(x) = 7\). This means the derivative of the function \(f(x) = 7x - 3\) is 7.

Key concepts

Difference Quotient

The concept of the difference quotient is foundational in calculus, particularly when we are on the quest to find the derivative of a function. Imagine it as a way to measure how a function’s output changes, or 'differs', as its input changes slightly. When we calculate the difference quotient, we take the function value at a point, say x, and compare it to the function value at a point very close to x, which is x + h. The formula looks like this: \[\frac{f(x + h) - f(x)}{h}\]This ratio gives us an average rate of change between the two points. By subsequently letting h shrink closer and closer to zero, we refine our average to an instantaneous rate of change at the point x, which is the derivative we’re seeking. The exercise at hand requires that we use this method to determine the derivative, thus deepening understanding of how derivatives reflect the behavior of functions.

Limit Definition of Derivative

The limit definition of the derivative is a critical concept that actually gives birth to the idea of a derivative. To get the derivative of a function at a specific point, we apply the limit to the difference quotient as h approaches zero. The beauty of this method lies in its ability to transition from an average rate of change to the exact instant where the change is occurring, hence capturing the essence of 'instantaneous rate of change'. Mathematically, the derivative f'(x) can be written as: \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]In the given exercise, applying this limit definition is how we accurately find the slope of the tangent line to the function at any point x. This process is essential not only for academic practice but also for applications in physics, economics, and other sciences where understanding change is crucial.

Calculus Exercises

Calculus exercises serve as a practical ground for students to apply theoretical concepts and solidify their understanding. These exercises challenge students to engage with key operations of calculus, like finding derivatives, and ensure that students can not only repeat a formula but also understand the significance of each step in the process. For instance, our example exercise involves finding the derivative via the limit of the difference quotient, animproving understanding of how algebraic simplification leads to identifying the rate at which a function is changing at any given point. Such exercises fortify the skills needed for more advanced topics in calculus, modeling real-world problems, and even in predicting and analyzing changes in various fields.

Simplifying Expressions in Calculus

An important skill in calculus is simplifying expressions, which often includes expanding, factoring, and canceling terms to make the problem more approachable or to reveal its core components. As we've seen in the provided exercise, simplifying the numerator of the difference quotient is a crucial step towards finding the derivative. After the initial substitution, the resulting expression \[\frac{(7(x+h) - 3) - (7x - 3)}{h}\]consisted of a difference between two similar terms, which simplified down to \[\frac{7h}{h}\]Upon canceling the h terms, we reached the conclusion that the derivative is constant at 7. This demonstrates not just a key algebraic technique in calculus, but also that simplifying an expression can sometimes lead to immediate insights about the function's behavior without having to undergo complex calculations.