Problem 21 - Precalculus: Mathematics for Calculus

Question

Find the derivative of the function at the given number. $$f(x)=x-3 x^{2}, \quad \text { at }-1$$

Step-by-Step Solution

Answer

The derivative at \( x = -1 \) is 7.

Step 1: Identify the Function

The given function is \( f(x) = x - 3x^2 \). We are asked to find its derivative at the point \( x = -1 \).

Step 2: Differentiate the Function

Find the derivative \( f'(x) \) by applying the power rule. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). For \( f(x) = x - 3x^2 \), the derivative is:\[f'(x) = \frac{d}{dx}(x) - \frac{d}{dx}(3x^2).\]The derivative of \( x \) is 1, and the derivative of \( 3x^2 \) is \( 6x \). So,\[ f'(x) = 1 - 6x.\]

Step 3: Evaluate the Derivative at the Given Point

Substitute \( x = -1 \) into the derivative \( f'(x) = 1 - 6x \):\[f'(-1) = 1 - 6(-1).\]Calculate this:\[ 1 - 6(-1) = 1 + 6 = 7.\]

Step 4: Conclusion

The derivative of the function \( f(x) = x - 3x^2 \) at \( x = -1 \) is \( f'(-1) = 7 \).

Key Concepts

Power Rule

Understanding how to use the power rule is essential in derivative calculation. In the expression of the power rule, if you have a variable to a power, such as \(x^n\), the derivative is calculated as \(n \cdot x^{n-1}\). This process reduces the power by one and multiplies the variable by the original power.
It's a straightforward and widely applicable tool in calculus, allowing you to calculate derivatives for polynomial terms quickly. For example, for a term like \(3x^2\), using the power rule, you bring down the 2, and multiply it by the coefficient (3) to get \(6x\). This results in the derivative being \(6x\).
Using the power rule efficiently can make more complex derivative computations much easier. We always apply it to each term separately in a function, and then combine them to find the full derivative.

Function Evaluation

Function evaluation becomes particularly important when you need to find specific values after differentiation. It involves substituting a particular value, such as \(x = -1\), into your derivative function.
After obtaining the derivative \(f'(x)\), as \(1 - 6x\) in this case, you simply replace \(x\) with \(-1\). This means you calculate \(f'(-1) = 1 - 6(-1)\), which simplifies to \(7\).
This process helps us find the rate of change or the slope of the tangent line at that specific point, which is crucial in understanding the behavior of functions at particular points in their domain.

Derivative Calculation

Calculating derivatives is one of the fundamental tasks in calculus, often used to investigate how a function behaves. The derivative represents the rate of change of a function's output with respect to its input.
In this example, we first took the function \(f(x) = x - 3x^2\) and calculated the derivative using the power rule. The step-by-step process involved differentiating each term independently to get a new function, \(f'(x) = 1 - 6x\).
The ease of derivative calculation increases with practice, making the process faster and more intuitive over time. Understanding how small changes in input affect the output is a powerful tool in various fields, from physics to economics.

Differentiation

Differentiation is a central concept in calculus, allowing us to determine how functions change over small intervals. It essentially answers questions about instantaneous rates of change.
In our exercise, by differentiating \(f(x) = x - 3x^2\), we were able to derive \(f'(x) = 1 - 6x\). It shows how this quadratic function behaves as \(x\) changes, particularly useful in predicting trends.
Differentiation is used across many disciplines to model and solve real-world problems involving rates, such as speed, acceleration, and optimization. By mastering differentiation techniques, students can apply these skills to a wide array of mathematical and practical challenges.