Problem 57 - Calculus

Question

Find the derivatives in algebraically.$$f(x)=5 x^{2} \text { at } x=10$$

Step-by-Step Solution

Answer

The derivative at \( x = 10 \) is 100.

Step 1: Identify the function

The given function is \( f(x) = 5x^2 \). We want to find its derivative algebraically, and evaluate it at \( x = 10 \).

Step 2: Differentiate the function

To find the derivative of \( f(x) = 5x^2 \), apply the power rule for differentiation. The power rule states that \( \frac{d}{dx}[ax^n] = nax^{n-1} \). Therefore:\[ f'(x) = \frac{d}{dx}[5x^2] = 2 \cdot 5x^{2-1} = 10x \].

Step 3: Evaluate the derivative at the given point

Substitute \( x = 10 \) into the derivative \( f'(x) = 10x \) to find the slope at this point:\[ f'(10) = 10 \cdot 10 = 100 \].

Step 4: Interpret the result

The value \( f'(10) = 100 \) represents the rate of change of the function \( f(x) = 5x^2 \) at \( x = 10 \). This means the slope of the tangent to the curve at \( x = 10 \) is 100.

Key Concepts

Understanding the Power Rule in Calculus

The power rule is a fundamental method in calculus for finding the derivative of functions. It is particularly useful when working with polynomials. The rule states that if you have a function of the form \( ax^n \), its derivative is given by \( n \, ax^{n-1} \). This means you multiply the coefficient \( a \) by the exponent \( n \), then reduce the exponent by one.
For example, in the function \( f(x) = 5x^2 \):

• Identify \( a = 5 \) and \( n = 2 \).
• Apply the power rule: multiply 5 by 2 to get 10.
• Reduce the power of \( x \) from 2 to 1, resulting in \( f'(x) = 10x \).

The power rule simplifies the process of differentiation, making it quick and efficient to find the rate of change of polynomial functions.

Derivative Evaluation Explained

Evaluating a derivative means finding the value of the derivative function at a specific point. This involves substituting a given value of \( x \) into the derivative. After applying the power rule to our original function \( f(x) = 5x^2 \), we derived \( f'(x) = 10x \). To evaluate this at \( x = 10 \), we substitute 10 into the derivative:

• \( f'(10) = 10 \times 10 \)
• This simplifies to \( f'(10) = 100 \).

Evaluation is essential for understanding how a function behaves at specific points. In this case, it tells us how steep the slope of the tangent line to the function is at \( x = 10 \). This slope value is a precise numeric expression of the rate of change at that point.

Understanding Rate of Change in Context

The rate of change is a crucial concept in understanding how a function's output changes relative to its input. In calculus, the derivative represents this rate. It tells you how fast or slow a function is changing at any point. For the function \( f(x) = 5x^2 \), the derivative \( f'(x) = 10x \) describes the rate of change.
When we evaluated \( f'(10) = 100 \), it indicated that the function was increasing at a rate of 100 units of \( y \) for every unit increase in \( x \) around \( x = 10 \).
The concept of rate of change is widely applicable:

• Interpreting how quickly a car accelerates based on velocity functions.
• Analyzing population growth rates by evaluating models at specific times.
• Determining economic trends through derivative evaluation of profit functions.

Understanding this concept is key to applying calculus in real-world scenarios and interpreting various dynamic systems. Embracing it helps in visualizing and predicting changes efficiently.