Question

Use the definition of the derivative to compute the derivative of the given function. $$f(x)=2 x$$

Short answer

The derivative of \( f(x)=2x \) is 2.

Step-by-step solution

Recall the Definition of Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as the limit \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). This represents the function's rate of change at that point.

Substitute into the Derivative Definition

Substitute the function \( f(x) = 2x \) into the definition of the derivative. We get: \( \lim_{h \to 0} \frac{2(x+h) - 2x}{h} \).

Simplify the Expression

Simplify the expression inside the limit: \( 2(x+h) - 2x = 2x + 2h - 2x = 2h \). Thus, the expression becomes \( \lim_{h \to 0} \frac{2h}{h} \).

Cancel Terms and Evaluate the Limit

Cancel the \( h \) terms: \( \frac{2h}{h} = 2 \). Since there are no \( h \) terms remaining, the limit is straightforward: \( \lim_{h \to 0} 2 = 2 \).

State the Result

Therefore, the derivative of the function \( f(x) = 2x \) is \( 2 \). This means the function increases at a constant rate of 2.

Key concepts

Definition of Derivative

In calculus, the concept of a derivative is fundamental. The derivative of a function provides us with valuable information about how the function behaves at any given point. More precisely, it describes the function's instantaneous rate of change at a specific point on its curve. This is akin to finding out the speed of a car at an exact moment, rather than over a period of time.

The formal definition of a derivative, often captured algebraically, is given by the limit process:

• \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This formula essentially computes the slope of the tangent line at a point on the function's graph. Here, \( h \) is a tiny increment in \( x \), and the limit tells us what happens as \( h \) shrinks closer and closer to zero.

Importantly, a derivative's existence depends on this limit being finite and existing for the given point on the function. It shows us how sensitive the function is to changes in \( x \). For students, grasping the definition is crucial as it forms the basis for further calculus concepts.

Limits

The notion of limits is indispensable when discussing derivatives, as they provide the mathematical foundation for defining derivatives. A limit helps us understand the behavior of functions as they curve or approach a specific value.

When we calculate a derivative using its definition, we rely on the limit:

• \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This limit reveals what happens to the expression \( \frac{f(x+h) - f(x)}{h} \) as \( h \), which symbolizes a very small change in \( x \), approaches zero. In simpler terms, limits help in determining the value that a function approaches as the input gets infinitely close to some point.

Understanding limits is the key to unlocking the powerful techniques in calculus, including derivatives and integrals. It's like peeking at what a graph does just by getting super close to a point without actually reaching it. So, in the case of derivative, we are interested in how the slope of secant lines (lines connecting two points on the graph) turns into the slope of the tangent (a line just touching the graph at that point).

Mastering limits might take some time, but it will greatly enhance how well one understands calculus.

Rate of Change

The concept of 'rate of change' is at the heart of derivatives. When we take the derivative of a function, we essentially calculate its rate of change. This rate of change tells us how fast or slow one quantity changes as another quantity changes. For example, in physics, velocity is a rate of change – it tells us how position changes with time.

With our given function \( f(x) = 2x \), to find its derivative, we found that the derivative is \( 2 \). This result interprets into the rate of change being constant at 2 for any value of \( x \).

• If \( f(x) \) was a linear function - which it is in our case - this rate of change is simply the slope of the line.
• The rate of change signifies how much \( f(x) \) increases for every unit increase in \( x \).

This is straightforward for linear functions, but the power of derivatives shines when dealing with non-linear, more complex functions where the rate of change isn't constant.

Understanding rate of change supports numerous real-life applications, from predicting economic trends to optimizing solutions in engineering. Once you grasp this, you'll see that derivatives not only solve mathematical problems but also explain many natural phenomena.