Problem 107 - Calculus Volume 1

Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=5 x^{3}-x+1 $$

Step-by-Step Solution

Answer

\( f'(x) = 15x^2 - 1 \)

Step 1: Identify the Function

The given function is \( f(x) = 5x^3 - x + 1 \). This is a polynomial function of degree 3, where each term can be differentiated separately.

Step 2: Apply the Power Rule to Each Term

The power rule for differentiation states that \( \frac{d}{dx} [x^n] = nx^{n-1} \). We will apply this rule to each term of the function separately. The first term is \( 5x^3 \), the second term is \( -x \), and the last term is \( +1 \).

Step 3: Differentiate the First Term

Differentiate \( 5x^3 \) using the power rule: \( \frac{d}{dx} [5x^3] = 3 \cdot 5x^{3-1} = 15x^2 \).

Step 4: Differentiate the Second Term

Differentiate \( -x \) using the power rule: \( \frac{d}{dx} [-x] = -1 \cdot x^{1-1} = -1 \).

Step 5: Differentiate the Constant Term

The derivative of a constant is 0. Therefore, \( \frac{d}{dx} [1] = 0 \).

Step 6: Combine the Derivatives

Combine the derivatives of each term to find \( f'(x) \): \( f'(x) = 15x^2 - 1 + 0 \). Hence, \( f'(x) = 15x^2 - 1 \).

Key Concepts

polynomial function

A polynomial function is an expression made up of variables and constants combined using addition, subtraction, and multiplication. In general, a polynomial can be written as:

• The sum of terms, each consisting of a constant (called a coefficient) multiplied by a whole number power of a variable.
• For instance, the expression \( f(x) = 5x^3 - x + 1 \) is a polynomial function.
• Here, the coefficients are 5, -1, and 1, while the variable is \( x \).
• The degree of the polynomial is determined by the term with the highest exponent, which in this case is 3 (from \( 5x^3 \)).

Understanding polynomial functions is crucial because they appear often in equations that model real-life situations. They can represent anything from the trajectory of a ball to changes in stock prices over time.

power rule

The power rule is a fundamental tool in differentiation. It allows us to easily find the derivative of terms that are powers of a variable.To apply the power rule:

• If you have a term \( x^n \), the derivative is \( nx^{n-1} \).
• This means you multiply by the power and then subtract one from the exponent.

In our example:

• For \( 5x^3 \), differentiate to get \( 15x^2 \), since \( 3 \times 5 = 15 \) and \( 3-1 = 2 \).
• For \( -x \), rethink of it as \( -1x^1 \). Using the power rule, it becomes \( -1 \cdot 1x^{1-1} = -1 \).
• Constants, like \( +1 \), have a derivative of 0 because constants do not change as \( x \) changes.

The power rule simplifies differentiation, especially for polynomial functions, making complex calculus problems more manageable.

derivative

A derivative is a concept that measures how a function changes as its input changes. Think of it as the function's sensitivity to change.The derivative, denoted as \( f'(x) \), tells you the rate of change or slope of a function at any given point.

• For instance, the derivative of \( f(x) = 5x^3 - x + 1 \) after applying the power rule is \( f'(x) = 15x^2 - 1 \).
• Here, \( 15x^2 \) represents how the cubic term influences the rate of change, while \(-1\) reflects the linear component's effect.

Derivatives are crucial in fields such as physics, engineering, and economics:

• They help in understanding the behavior of systems over time.
• From predicting how stock prices change to figuring out the speed of an object.

Hence, mastering derivatives is essential for solving real-world problems efficiently.