Problem 22 - Precalculus: Mathematics for Calculus

Question

Find the derivative of the function at the given number. $$f(x)=x+x^{3}, \quad \text { at } 1$$

Step-by-Step Solution

Answer

The derivative at \( x = 1 \) is 4.

Step 1: Write down the function

Start with the given function: \( f(x) = x + x^3 \).

Step 2: Differentiate the function with respect to x

To find the derivative \( f'(x) \), apply the power rule \( \frac{d}{dx} x^n = n\cdot x^{n-1} \) to each term:- The derivative of \( x \) is \( 1 \).- The derivative of \( x^3 \) is \( 3x^2 \).Combine these results to find:\[f'(x) = 1 + 3x^2\]

Step 3: Evaluate the derivative at the given number

Now that we have \( f'(x) = 1 + 3x^2 \), substitute \( x = 1 \) into the derivative:\[f'(1) = 1 + 3(1)^2 = 1 + 3 = 4\]So, the derivative of the function at \( x = 1 \) is \( 4 \).

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus for finding the derivative of polynomial functions quickly. When dealing with any term of the form \( x^n \), where \( n \) is a real number, the power rule states that the derivative is \( n \cdot x^{n-1} \). This means we multiply the exponent \( n \) by the coefficient of \( x \) and then decrease the exponent by one.

In the case of our exercise, we applied the power rule to both terms of the function \( f(x) = x + x^3 \). For the first term \( x \), which can be seen as \( x^1 \), applying the power rule results in \( 1 \cdot x^0 = 1 \). This is because the exponent \( n = 1 \) diminishes to zero, turning \( x^0 \) into 1. For the second term, \( x^3 \) applies the power rule to yield \( 3 \cdot x^2 \), as the exponent decreases from 3 to 2.

Differentiation

Differentiation is the process of finding the derivative of a function, which describes how the function's value changes as its input changes. This is a critical concept in calculus that allows us to understand the behavior of graphs, optimize functions, and solve complex real-world problems.

In our example, we started with a function \( f(x) = x + x^3 \). Differentiation involves taking the derivative of each term separately. Using the power rule, we found the derivative of the entire function to be\( f'(x) = 1 + 3x^2 \). Differentiation is the step-by-step transformation of a function from its original form to its rate of change, utilizing rules like the power rule to execute these transformations proficiently.

The derivative is an essential aspect as it provides insights on the slope of the tangent line to the function at any point. This helps in understanding how fast or slow a function changes, which is valuable in fields such as physics, engineering, and economics.

Evaluation of Derivatives

Once we have found the derivative of a function, the next step is to evaluate it at a specific point. Evaluating a derivative at a certain value tells us the exact rate of change or the slope of the function at that point.

In our exercise, after finding the derivative \( f'(x) = 1 + 3x^2 \), we were asked to evaluate \( f'(x) \) at \( x = 1 \). This means substituting \( x = 1 \) into the derivative equation: \( f'(1) = 1 + 3(1)^2 \), resulting in \( f'(1) = 4 \).

Thus, the slope of the function \( f(x) = x + x^3 \) at the point \( x = 1 \) is 4. This step is crucial in many applications such as finding the gradient of a curve at a point or determining the instantaneous rate of change in various scientific contexts.