Problem 17 - Calculus Single Variable

Question

A function \(f\) and a point \(c\) are given. Calculate \(f^{\prime}(c)\). $$ f(x)=5 x^{2}-21 x \quad c=3 $$

Step-by-Step Solution

Answer

The derivative \( f^{\prime}(3) \) is 9.

Step 1: Understand the Definition of Derivative

The derivative of a function at a point gives the slope of the tangent line to the function's graph at that particular point. Mathematically, the derivative of a function \( f(x) \) with respect to \( x \) at some point \( c \) is denoted as \( f^{ ext{'} }(c) \).

Step 2: Differentiate the Function

To find the derivative of \( f(x) = 5x^2 - 21x \), we differentiate each term separately. The derivative of \( 5x^2 \) is \( 2 \cdot 5x^{2-1} = 10x \) and the derivative of \( -21x \) is \( -21 \). Therefore, the derivative of \( f(x) \) is \( f^{ ext{'}}(x) = 10x - 21 \).

Step 3: Evaluate the Derivative at the Given Point

Now, we substitute \( x = c = 3 \) into the derivative \( f^{'}(x) = 10x - 21 \). This gives us \( f^{ ext{'}}(3) = 10(3) - 21 = 30 - 21 = 9 \).

Key Concepts

Derivative at a point

Understanding the derivative at a point is a crucial part of calculus. The derivative represents the rate of change of a function concerning its variable, like a snapshot of the function's behavior at a particular moment.
For any function \( f(x) \), its derivative at a specific point \( c \), denoted as \( f'(c) \), tells us how the function is changing at that precise location.

• You can think of the derivative as the slope of a line that just touches the curve at a single point, known as the tangent line.
• The process of differentiation involves finding this rate of change.

In the exercise above, the function \( f(x) = 5x^2 - 21x \) is differentiated to find \( f'(c) \) at \( c = 3 \). By substituting \( x = 3 \) into the differentiated function \( 10x - 21 \), we find that \( f'(3) = 9 \).
This means the function is increasing at a rate of 9 units at \( x = 3 \).

Slope of tangent line

The slope of the tangent line at a specific point on a graph of a function is synonymous with the derivative of the function at that point.
This slope indicates how steep the graph is at that particular location and in what direction it is moving. When we say that the slope of the tangent line is positive, it means the graph is rising as you move from left to right.

• If the slope is negative, the graph is falling in the same direction.
• A zero slope implies that the graph is flat at that point.

In our specific problem, we computed the slope of the tangent line to the function \( f(x) = 5x^2 - 21x \) at \( x = 3 \).
By calculating \( f'(3) = 9 \), we found the slope of the tangent line is 9. This positive value suggests that the graph is increasing steeply at this point.

Polynomial differentiation

Polynomial differentiation involves applying simple rules to differentiate polynomial functions. Each term in a polynomial can be differentiated individually, using the power rule.
The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \).

• This rule allows us to tackle each term separately and systematically to find the overall derivative.
• For instance, in \( f(x) = 5x^2 - 21x \), we apply the rule to each term:

- For \( 5x^2 \), differentiate using the power rule to get \( 2 \times 5x^{2-1} = 10x \).
- For \( -21x \), the derivative is simply \( -21 \), as the exponent of \( x \) is 1.
Combining these results gives \( f'(x) = 10x - 21 \). This result shows how the function is continuously changing across different values of \( x \).
Polynomial differentiation helps us analyze and predict the behavior and trends of polynomial functions efficiently.