Problem 7
Question
Find the derivative of the given function. $$ F(x)=\sqrt[3]{2 x^{3}-5 x^{2}+x} $$
Step-by-Step Solution
Verified Answer
F'(x) = \( \frac{1}{3} (2x^3 - 5x^2 + x)^{-2/3} \cdot (6x^2 - 10x + 1) \).
1Step 1: Rewrite the Function
Rewrite the function in a form that is easier to differentiate. Recall that the cube root is equivalent to raising to the power of \( \frac{1}{3} \). Thus, the function can be rewritten as \( F(x) = (2x^3 - 5x^2 + x)^{1/3} \).
2Step 2: Apply the Chain Rule
Use the chain rule to differentiate \( F(x) \). The chain rule states that if \( y = u^n \), then \( \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \). Let \( u = 2x^3 - 5x^2 + x \) and \( n = \frac{1}{3} \).
3Step 3: Differentiate the Outer Function
Differentiate the outer function \( u^{1/3} \). Using the chain rule, we get \( \frac{d}{dx} (u^{1/3}) = \frac{1}{3} u^{-2/3} \). Substituting back, we have: \( F'(x) = \frac{1}{3} (2x^3 - 5x^2 + x)^{-2/3} \).
4Step 4: Differentiate the Inner Function
Differentiate the inner function \( u = 2x^3 - 5x^2 + x \). We get \( \frac{du}{dx} = 6x^2 - 10x + 1 \).
5Step 5: Combine the Results
Combine the results from steps 3 and 4 to obtain the final derivative. Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: \( F'(x) = \frac{1}{3} (2x^3 - 5x^2 + x)^{-2/3} \cdot (6x^2 - 10x + 1) \).
Key Concepts
derivative calculationchain ruledifferentiation techniquesinner and outer functions
derivative calculation
Understanding how to calculate the derivative is essential for finding rates of change. For the function given, the process involves multiple steps streamlined into a methodical approach. Calculating the derivative involves differentiating a function, which means finding its rate of change at any point.
The derivative of a function provides insight into its behavior - how it increases, decreases, or remains constant.
In our example, we start by rewriting the function in a more workable form: \( F(x) = (2x^3 - 5x^2 + x)^{1/3} \). This step is crucial because it simplifies the differentiation process.
By applying differentiation rules, specifically the chain rule, we derive the inner and outer components. Finally, we multiply these results to find the derivative.
The derivative of a function provides insight into its behavior - how it increases, decreases, or remains constant.
In our example, we start by rewriting the function in a more workable form: \( F(x) = (2x^3 - 5x^2 + x)^{1/3} \). This step is crucial because it simplifies the differentiation process.
By applying differentiation rules, specifically the chain rule, we derive the inner and outer components. Finally, we multiply these results to find the derivative.
chain rule
The chain rule is a powerful differentiation technique that allows us to differentiate composite functions. A composite function is one that combines two or more functions.
The chain rule formula is given by: \ \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \), where \( y \) is a function of \( u \) and \( u \) is a function of \( x \).
Essentially, we first find the derivative of the outer function concerning the inner function and then multiply it by the derivative of the inner function relating to \( x \).
In our solution, we let \( u = 2x^3 - 5x^2 + x \) and differentiated the outer function: \( (u^{1/3}) \), resulting in \( \frac{1}{3} u^{-2/3} \). We then multiplied this by the derivative of the inner function, \(6x^2 - 10x + 1 \).
The chain rule formula is given by: \ \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \), where \( y \) is a function of \( u \) and \( u \) is a function of \( x \).
Essentially, we first find the derivative of the outer function concerning the inner function and then multiply it by the derivative of the inner function relating to \( x \).
In our solution, we let \( u = 2x^3 - 5x^2 + x \) and differentiated the outer function: \( (u^{1/3}) \), resulting in \( \frac{1}{3} u^{-2/3} \). We then multiplied this by the derivative of the inner function, \(6x^2 - 10x + 1 \).
differentiation techniques
Differentiation techniques vary and can be specific to the type and structure of the function. Basic techniques include applying the power rule, product rule, quotient rule, and chain rule.
The power rule: States that the derivative of \( x^n \) is \( nx^{n-1} \).
Product rule: Used when dealing with products of two or more functions: \( (fg)' = f'g + fg' \).
Quotient rule: Used for quotients, given by \( (f/g)' = (f'g - fg')/g^2 \).
For our function, we applied the chain rule, which is particularly effective for nested functions or those raised to a power.
Differentiating each part correctly and then combining them, as shown in our solution, is key to mastering these techniques.
The power rule: States that the derivative of \( x^n \) is \( nx^{n-1} \).
Product rule: Used when dealing with products of two or more functions: \( (fg)' = f'g + fg' \).
Quotient rule: Used for quotients, given by \( (f/g)' = (f'g - fg')/g^2 \).
For our function, we applied the chain rule, which is particularly effective for nested functions or those raised to a power.
Differentiating each part correctly and then combining them, as shown in our solution, is key to mastering these techniques.
inner and outer functions
Identifying inner and outer functions is an essential step in applying the chain rule effectively. An inner function is the one inside another function, and the outer function is the one applied to it.
For instance, if we have a composite function \( F(x) = \big( g(h(x)) \big) \), then \( h(x) \) is the inner function and \( g(u) \) where \( u = h(x) \), is the outer function.
In our example, the inner function is \( u = 2x^3 - 5x^2 + x \) and the outer function is \( u^{1/3} \). To differentiate, we:
1. Differentiate the outer function with respect to the inner function: \( \frac{1}{3} u^{-2/3} \)).
2. Multiply this by the derivative of the inner function: \( 6x^2 - 10x + 1 \).
The combination of these derivatives gives us the complete derivative of the composite function.
For instance, if we have a composite function \( F(x) = \big( g(h(x)) \big) \), then \( h(x) \) is the inner function and \( g(u) \) where \( u = h(x) \), is the outer function.
In our example, the inner function is \( u = 2x^3 - 5x^2 + x \) and the outer function is \( u^{1/3} \). To differentiate, we:
1. Differentiate the outer function with respect to the inner function: \( \frac{1}{3} u^{-2/3} \)).
2. Multiply this by the derivative of the inner function: \( 6x^2 - 10x + 1 \).
The combination of these derivatives gives us the complete derivative of the composite function.
Other exercises in this chapter
Problem 7
Differentiate the given function by applying the theorems of this section. $$ v(r)=\frac{4}{3} \pi r^{3} $$
View solution Problem 7
Find the derivative of the given function. $$ h(u)=\left(3 u^{2}+5\right)^{3}(3 u-1)^{2} $$
View solution Problem 7
Find \(D_{x} y\) by implicit differentiation. $$ \sqrt{x}+\sqrt{y}=4 $$
View solution Problem 7
The profit of a retail store is \(100 y\) dollars when \(x\) dollars are spent daily on advertising and \(y=2500+36 x-0.2 x^{2}\). Use the derivative to determi
View solution