Problem 15 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=a x^{3}+b x^{2}+c x+d \quad(a, b, c, d\text { constant) } $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 3ax^2 + 2bx + c \).

Step 1: Understand the Problem

We need to find the derivative of the function \( f(x) = ax^3 + bx^2 + cx + d \). This means we have to apply the rules of differentiation to each term in the polynomial.

Step 2: Differentiate the First Term

Differentiate the first term \( ax^3 \). Using the power rule of differentiation \( \frac{d}{dx}[x^n] = nx^{n-1} \), the derivative of \( ax^3 \) is \( 3ax^2 \).

Step 3: Differentiate the Second Term

Differentiate the second term \( bx^2 \). Applying the power rule again, the derivative is \( 2bx \).

Step 4: Differentiate the Third Term

Differentiate the third term \( cx \). The derivative of \( cx \) using the power rule (or noting \( x^1 \) becomes \( 1x^0 = 1 \)) is \( c \).

Step 5: Differentiate the Constant Term

Constant terms disappear when differentiated since the derivative of a constant \( d \) is 0.

Step 6: Combine the Derivatives

Combine all the derivatives found in the previous steps to obtain \( f'(x) = 3ax^2 + 2bx + c \).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus that simplifies the process of differentiation. It states that when you have a term in the form of \( x^n \), its derivative is \( nx^{n-1} \). This rule helps us find the slope of tangent lines to polynomials and is a building block for more complex calculus problems.

Let's apply the power rule to an example: if you have \( f(x) = x^3 \), according to the power rule, the derivative \( f'(x) \) is \( 3x^2 \). Notice how we take the power (3) and multiply it by the coefficient (1 in this case), then reduce the exponent by one. This approach can be applied to any term of a polynomial, making the process both straightforward and quick.

Polynomial Differentiation

Polynomial differentiation involves finding the derivative of a polynomial function. A polynomial is simply a sum of terms, each consisting of a variable raised to a power and multiplied by a coefficient.

In the context of the exercise \( f(x)=ax^3+bx^2+cx+d \), each term is dealt with separately using the power rule. This means:

• For \( ax^3 \), the derivative is \( 3ax^2 \).
• For \( bx^2 \), the derivative is \( 2bx \).
• For \( cx \), the derivative is \( c \), since \( x^1 \)'s derivative follows from reducing the power to zero, leaving just the coefficient.
• For the constant \( d \), the derivative is 0 because constants do not change as \( x \) changes.

This step-by-step application of the power rule to each component of a polynomial lets us derive the overall function, capturing every rate of change embedded within it.

Step-by-Step Differentiation

Step-by-step differentiation is a method where we break down the process of finding a derivative into manageable sections. This ensures that each term of the function is differentiated correctly and that no steps are overlooked.

In the given problem, finding \( f'(x) \) involves multiple steps:

• Identify the Function: Understand what each term of \( f(x)=ax^3+bx^2+cx+d \) represents.
• Apply Power Rule: Use the rule to differentiate each term independently.
• Combine the Derivatives: Sum up all the differentiated parts to arrive at the derivative function.

This systematic approach provides clarity and accuracy, especially when applied to complex functions. By following these steps, you can confidently handle any polynomial differentiation problem.