Problem 20
Question
Differentiate the given function by applying the theorems of this section. $$ f(x)=\left(x^{2}-3 x+2\right)\left(2 x^{3}+1\right) $$
Step-by-Step Solution
Verified Answer
The derivative is: \[ f'(x) = 10x^4 - 24x^3 + 12x^2 + 2x - 3 \]
1Step 1: Identify Function Components
The given function is a product of two functions, so identify them as: \[ u(x) = x^2 - 3x + 2 \] \[ v(x) = 2x^3 + 1 \]
2Step 2: Apply the Product Rule
Use the product rule for differentiation which states: \[ (u \times v)' = u'v + uv' \]
3Step 3: Differentiate Each Function
Differentiate both functions separately: \[ u'(x) = \frac{d}{dx}(x^2 - 3x + 2) = 2x - 3 \] \[ v'(x) = \frac{d}{dx}(2x^3 + 1) = 6x^2 \]
4Step 4: Substitute Back into the Product Rule Formula
Insert the differentiated functions into the product rule formula: \[ (x^2 - 3x + 2)(6x^2) + (2x - 3)(2x^3 + 1) \]
5Step 5: Simplify the Expression
Expand and simplify the expression: \[ (x^2 - 3x + 2)(6x^2) = 6x^4 - 18x^3 + 12x^2 \] \[ (2x - 3)(2x^3 + 1) = 4x^4 - 6x^3 + 2x - 3 \] Combine them to get: \[ f'(x) = 6x^4 - 18x^3 + 12x^2 + 4x^4 - 6x^3 + 2x - 3 \] Combine like terms: \[ f'(x) = 10x^4 - 24x^3 + 12x^2 + 2x - 3 \]
Key Concepts
product rule for differentiationderivative of polynomial functionsstep-by-step differentiation
product rule for differentiation
The product rule is an essential theorem in calculus that helps in differentiating products of two functions. When you have a function that is the product of two other functions, like \(f(x) = u(x) * v(x) \), you use the product rule.
The product rule states: \((u \times v)' = u'v + uv'\).
This means you first differentiate the first function \(u'\), then multiply it by the second function \(v\) and add the result to the product of the first function \(u\) and the derivative of the second function \(v'\).
Here's a simple way to remember it: different parts of the product get their derivative taken separately, summed together in a specific order.
The product rule states: \((u \times v)' = u'v + uv'\).
This means you first differentiate the first function \(u'\), then multiply it by the second function \(v\) and add the result to the product of the first function \(u\) and the derivative of the second function \(v'\).
Here's a simple way to remember it: different parts of the product get their derivative taken separately, summed together in a specific order.
derivative of polynomial functions
Polynomials are expressions made up of variables and constants combined using addition, subtraction, and multiplication.
The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).
Differentiating polynomials involves applying rules of differentiation to each term separately. The power rule is most often used here, which states: if \(f(x) = ax^n\), then \(f'(x) = anx^{n-1}\).
For example, differentiating \(x^2\) gives \(2x\), and differentiating \(-3x\) gives \(-3\).
This rule simplifies the process considerably, making it easy to handle polynomial equations in calculus.
The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).
Differentiating polynomials involves applying rules of differentiation to each term separately. The power rule is most often used here, which states: if \(f(x) = ax^n\), then \(f'(x) = anx^{n-1}\).
For example, differentiating \(x^2\) gives \(2x\), and differentiating \(-3x\) gives \(-3\).
This rule simplifies the process considerably, making it easy to handle polynomial equations in calculus.
step-by-step differentiation
Let's walk through the steps to differentiate a given function using the product rule.
Consider the function given by: \(f(x)=(x^2-3x+2)(2x^3+1)\).
Follow these steps to differentiate it:
Consider the function given by: \(f(x)=(x^2-3x+2)(2x^3+1)\).
Follow these steps to differentiate it:
- Step 1: Identify the components. Here, \(u(x) = x^2 - 3x + 2\) and \(v(x) = 2x^3 + 1\).
- Step 2: Apply the product rule: \((u \times v)' = u'v + uv'\).
- Step 3: Differentiate each function separately: \(u'(x) = 2x - 3\) and \(v'(x) = 6x^2\).
- Step 4: Substitute back into the product rule:\( (x^2 - 3x + 2)(6x^2) + (2x - 3)(2x^3 + 1) \).
- Step 5: Simplify the expression: \(6x^4 - 18x^3 + 12x^2 + 4x^4 - 6x^3 + 2x - 3\).
- Combine like terms to get the final derivative: \f'(x) = 10x^4 - 24x^3 + 12x^2 + 2x - 3\.
Other exercises in this chapter
Problem 20
Given \(b^{2} x^{2}-a^{2} y^{2}=a^{2} b^{2}\left(a\right.\) and \(b\) are constants), find \(D_{x}{\underline{\phantom{xx}}}^{2} y\) in simplest form.
View solution Problem 20
Find an equation of the normal line to the curve \(y=x^{3}-3 x\) that is parallel to the line \(2 x+18 y-9=0\).
View solution Problem 20
Find the derivative of the given function. $$ F(x)=\frac{(5 x-8)^{-2}}{\left(x^{2}+3\right)^{-3}} $$
View solution Problem 20
Find an equation of the tangent line to the curve \(y=(6-2 x)^{1 / 3}\) at each of the following points: \((-1,2),(1, \sqrt[3]{4})\), \((3,0),(5,-\sqrt[3]{4}),(
View solution