Problem 66
Question
Differentiate each function. \(f(x)=x\left(3 x^{3}+6 x-2\right)\left(3 x^{4}+7\right)\)
Step-by-Step Solution
Verified Answer
The derivative is \( f'(x) = 72x^7 + 108x^5 - 30x^4 + 84x^3 + 84x - 14 \).
1Step 1: Expand the Original Function
First, expand the function to make differentiation easier. The function is given as:\[ f(x) = x(3x^3 + 6x - 2)(3x^4 + 7) \]You must multiply the terms inside the parenthesis first then multiply by \(x\).
2Step 1.1: Multiply the Binomials
Expand \((3x^3 + 6x - 2)(3x^4 + 7)\). Use the distributive property to multiply each term:\[ (3x^3)(3x^4) + (3x^3)(7) + (6x)(3x^4) + (6x)(7) + (-2)(3x^4) + (-2)(7) \]Then, complete each multiplication.
3Step 1.2: Simplify the Expanded Expression
After performing the multiplication: - \(3x^3 \cdot 3x^4 = 9x^7\)- \(3x^3 \cdot 7 = 21x^3\)- \(6x \cdot 3x^4 = 18x^5\)- \(6x \cdot 7 = 42x\)- \(-2 \cdot 3x^4 = -6x^4\)- \(-2 \cdot 7 = -14\)The expression becomes:\[ 9x^7 + 18x^5 + 21x^3 - 6x^4 + 42x - 14 \]This is still inside the parentheses with an \(x\) multiplying in front, so we multiply through by \(x\):\[ f(x) = x(9x^7 + 18x^5 + 21x^3 - 6x^4 + 42x - 14) \]
4Step 1.3: Final Expansion Simplification
Distribute the \(x\):\[ f(x) = 9x^8 + 18x^6 + 21x^4 - 6x^5 + 42x^2 - 14x \]
5Step 2: Differentiate the Function
Differentiate term by term using the power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\):- Derivative of \(9x^8\) is \(72x^7\)- Derivative of \(18x^6\) is \(108x^5\)- Derivative of \(21x^4\) is \(84x^3\)- Derivative of \(-6x^5\) is \(-30x^4\)- Derivative of \(42x^2\) is \(84x\)- Derivative of \(-14x\) is \(-14\)
6Step 3: Write the Final Derivative
Combine the differentiated terms to get:\[ f'(x) = 72x^7 + 108x^5 - 30x^4 + 84x^3 + 84x - 14 \]
Key Concepts
Power RulePolynomial DifferentiationDistributive Property
Power Rule
When differentiating a polynomial, the power rule is one of the most straightforward and essential tools. It states that to differentiate a term of the form \(x^n\), you multiply the exponent \(n\) by the coefficient of the term and then reduce the exponent by one:\[ \frac{d}{dx}(x^n) = nx^{n-1} \]. This technique is applied individually to each term in the expression.
In the context of our exercise, when we encounter a term like \(9x^8\), the power rule allows us to differentiate it easily. Here, \(n = 8\), so the derivative is \(72x^7\), as we multiply 8 by the coefficient 9 and reduce the power by one.
This method simplifies the process of differentiation significantly and is especially useful for expressions with several polynomial terms. It demonstrates the beauty of calculus in breaking down complex expressions into manageable parts.
In the context of our exercise, when we encounter a term like \(9x^8\), the power rule allows us to differentiate it easily. Here, \(n = 8\), so the derivative is \(72x^7\), as we multiply 8 by the coefficient 9 and reduce the power by one.
This method simplifies the process of differentiation significantly and is especially useful for expressions with several polynomial terms. It demonstrates the beauty of calculus in breaking down complex expressions into manageable parts.
Polynomial Differentiation
Polynomial differentiation often involves applying the power rule to each term of a polynomial function.
A polynomial is a sum of terms, each composed of a coefficient and a variable raised to a non-negative integer power. Differentiating these individual terms results in a new polynomial with each exponent decreased by one. This new polynomial represents the rate of change of the original function.
In the exercise given, we expanded the original function to a polynomial and then calculated the derivative term by term. Each term's derivative was computed independently, yielding the final derivative \( f'(x) = 72x^7 + 108x^5 - 30x^4 + 84x^3 + 84x - 14 \).
Differentiating polynomials is a core skill in calculus, providing insights into how quantities change and relationships between different quantities.
A polynomial is a sum of terms, each composed of a coefficient and a variable raised to a non-negative integer power. Differentiating these individual terms results in a new polynomial with each exponent decreased by one. This new polynomial represents the rate of change of the original function.
In the exercise given, we expanded the original function to a polynomial and then calculated the derivative term by term. Each term's derivative was computed independently, yielding the final derivative \( f'(x) = 72x^7 + 108x^5 - 30x^4 + 84x^3 + 84x - 14 \).
Differentiating polynomials is a core skill in calculus, providing insights into how quantities change and relationships between different quantities.
Distributive Property
The distributive property is a cornerstone of algebra, allowing us to expand expressions by distributing multiplication over addition. This property is pivotal when simplifying expressions to make them amenable to differentiation.
For instance, in our exercise, we initially had \( f(x) = x(3x^3 + 6x - 2)(3x^4 + 7) \). To differentiate effectively, we first use the distributive property to expand \((3x^3 + 6x - 2)(3x^4 + 7)\) into a single polynomial. Each term inside the first parenthesis is multiplied by each term inside the second. The outcome is then simplified into a polynomial, ready for differentiation.
Once expanded, multiplying by \(x\) again uses the distributive property, making the expression fully expanded and suitable for applying the power rule. This approach underscores how essential the distributive property is in preparing functions for differentiation.
For instance, in our exercise, we initially had \( f(x) = x(3x^3 + 6x - 2)(3x^4 + 7) \). To differentiate effectively, we first use the distributive property to expand \((3x^3 + 6x - 2)(3x^4 + 7)\) into a single polynomial. Each term inside the first parenthesis is multiplied by each term inside the second. The outcome is then simplified into a polynomial, ready for differentiation.
Once expanded, multiplying by \(x\) again uses the distributive property, making the expression fully expanded and suitable for applying the power rule. This approach underscores how essential the distributive property is in preparing functions for differentiation.
Other exercises in this chapter
Problem 66
For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=3 x^{2}-5 x+4 $$
View solution Problem 66
For \(y=a x^{3}+b x^{2}+c x+d,\) find \(d^{3} y / d x^{3}\)
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Let \(f(u)=2 u^{5}\) and \(g(x)=u=\frac{3-x}{4+x}\) Find \((f \circ g)^{\prime}(-10)\)
View solution Problem 66
Is the function given by \(F(x)=\frac{1}{x^{2}-7 x+10}\) continuous at \(x=4 ?\) Why or why not?
View solution