Problem 2 Determine \(f^{\prime}(x)\) for ... [FREE SOLUTION]

Chapter 2: Problem 2

Determine \(f^{\prime}(x)\) for each of the following functions: a. \(f(x)=4 x-7\) b. \(f(x)=x^{3}-x^{2}\) c. \(f(x)=-x^{2}+5 x+8\) d. \(f(x)=\sqrt[3]{x}\) e. \(f(x)=\left(\frac{x}{2}\right)^{4}\) f. \(f(x)=x^{-3}\)

Short Answer

Expert verified

a. 4; b. 3x^2 - 2x; c. -2x + 5; d. \frac{1}{3}x^{-2/3}; e. \frac{x^3}{4}; f. -3x^{-4}.

Step by step solution

Differentiate Polynomial Function for a

The function given is \( f(x) = 4x - 7 \). To find the derivative \( f'(x) \), we apply the power rule. For a linear function \( ax + b \), the derivative is \( a \). Hence, the derivative of \( 4x \) is \( 4 \), and the derivative of the constant \( -7 \) is \( 0 \). Therefore, \( f'(x) = 4 \).

Differentiate Polynomial Function for b

For \( f(x) = x^3 - x^2 \), we apply the power rule \( nx^{n-1} \) to each term. The derivative of \( x^3 \) is \( 3x^2 \), and the derivative of \( x^2 \) is \( 2x \). Therefore, \( f'(x) = 3x^2 - 2x \).

Differentiate Quadratic Function for c

The function \( f(x) = -x^2 + 5x + 8 \) is a quadratic polynomial. Applying the power rule, the derivative of \( -x^2 \) is \( -2x \), the derivative of \( 5x \) is \( 5 \), and the derivative of the constant \( 8 \) is \( 0 \). Thus, \( f'(x) = -2x + 5 \).

Differentiate Cubic Root Function for d

For \( f(x) = \sqrt[3]{x} \), rewrite it as \( f(x) = x^{1/3} \). Use the power rule: the derivative is \( \frac{1}{3}x^{-2/3} \), simplifying to \( f'(x) = \frac{1}{3}x^{-2/3} \).

Differentiate Fractional Power Function for e

The function \( f(x) = \left( \frac{x}{2} \right)^4 \) can be rewritten as \( f(x) = \frac{x^4}{16} \). Applying the power rule, the derivative is \( \frac{4x^3}{16} = \frac{x^3}{4} \). Thus, \( f'(x) = \frac{x^3}{4} \).

Differentiate Negative Exponent Function for f

The function \( f(x) = x^{-3} \) requires the power rule. The derivative is \( -3x^{-4} \). Hence, \( f'(x) = -3x^{-4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule

The power rule is a fundamental principle in calculus when finding derivatives. It is used to differentiate terms of the form \( x^n \), where \( n \) is any real number. The power rule states that to differentiate \( x^n \), you multiply the exponent \( n \) by the term and then reduce the exponent by one. For example, the derivative of \( x^3 \) would be calculated as follows:

Multiply the current exponent (3) by the coefficient (1 in this case) to get 3.
Reduce the exponent by one to get the new exponent, which is 2.
The result is the derivative: \( 3x^2 \).

This method is quick and effective for differentiating polynomial terms, whether they are whole numbers or include fractions.

Polynomial Differentiation

Differentiating polynomials involves using the power rule across each term of the polynomial. A polynomial might look like \( x^3 - x^2 \) or \( -x^2 + 5x + 8 \). For each term:

Apply the power rule to find the derivative.
For constant numbers like -7 or 8, their derivative is zero, as constants have no rate of change.
Combine the derivatives of individual terms to get the overall derivative of the polynomial.

Let's see this with the polynomial \( x^3 - x^2 \):

The derivative of \( x^3 \) is \( 3x^2 \).
The derivative of \( x^2 \) is \( 2x \).
Thus, the derivative of the polynomial is \( 3x^2 - 2x \).

This systematic application of the power rule makes polynomial differentiation straightforward and clear.

Negative Exponent Differentiation

Negative exponents indicate terms flipped relative to the standard base position. For example, \( x^{-3} \) can be rewritten as \( \frac{1}{x^3} \). When differentiating terms with negative exponents, use the power rule:

Multiply the negative exponent by the base coefficient.
Subtract one from the negative exponent to get the new exponent.
For \( x^{-3} \), the derivative is \( -3x^{-4} \).

Negative exponents can seem tricky at first but follow the same principles of the power rule, making them easier to handle with practice.

Fractional Power Differentiation

When dealing with terms that have fractional exponents, such as \( x^{1/3} \), rewriting them can sometimes help clarify the differentiation process. For example, \( \sqrt[3]{x} \) can be rewritten as \( x^{1/3} \). In applying the power rule:

Multiply the fractional exponent by the base (which is often 1 if no other coefficient is specified).
Subtract one from the exponent, keeping the result as a fraction.
The derivative of \( x^{1/3} \) becomes \( \frac{1}{3}x^{-2/3} \).

Fractional powers require careful handling of fractions, but with consistent use of the power rule, they become much more manageable.

91影视

Determine \(f^{\prime}(x)\) for each of the following functions: a. \(f(x)=4 x-7\) b. \(f(x)=x^{3}-x^{2}\) c. \(f(x)=-x^{2}+5 x+8\) d. \(f(x)=\sqrt[3]{x}\) e. \(f(x)=\left(\frac{x}{2}\right)^{4}\) f. \(f(x)=x^{-3}\)

Short Answer

Step by step solution

Differentiate Polynomial Function for a

Differentiate Polynomial Function for b

Differentiate Quadratic Function for c

Differentiate Cubic Root Function for d

Differentiate Fractional Power Function for e

Differentiate Negative Exponent Function for f

Key Concepts

Power Rule

Polynomial Differentiation

Negative Exponent Differentiation

Fractional Power Differentiation

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