Problem 29
Question
Differentiate $$ f(x)=r s^{2} x^{3}-r x+s $$ with respect to \(x\). Assume that \(r\) and \(s\) are constants.
Step-by-Step Solution
Verified Answer
The derivative is \(3rs^2 x^2 - r\).
1Step 1: Differentiate each term separately
To differentiate the given function, we'll apply the power rule for each term separately. The function is given by: \[ f(x) = r s^2 x^3 - r x + s \] Differentiate each term with respect to \(x\): - \(rs^2 x^3\) - \(-rx\) - \(s\)
2Step 2: Apply the power rule to the first term
The power rule of differentiation states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Applying this rule to the first term, \(rs^2 x^3\): \[ \frac{d}{dx}(rs^2 x^3) = rs^2 \cdot 3x^{2} = 3rs^2 x^2 \].
3Step 3: Differentiate the second term
The term \(-rx\) can be differentiated using the constant multiple rule and the power rule: - The derivative of \(x\) with respect to \(x\) is \(1\). Thus, \(-r \cdot 1 = -r\).
4Step 4: Differentiate the third term
The term \(s\) is a constant with respect to \(x\), and the derivative of a constant is \(0\). Therefore, \[ \frac{d}{dx}(s) = 0 \].
5Step 5: Add the derivatives
Combine all the derived terms to find the derivative of the function. Adding all the differentiated parts: \[ \frac{d}{dx}[f(x)] = 3rs^2 x^2 - r + 0 \]. This simplifies to: \[ 3rs^2 x^2 - r \].
Key Concepts
Power RuleConstant Multiple RuleDerivative of a ConstantPolynomial Differentiation
Power Rule
The power rule is a fundamental technique in calculus that's used to differentiate functions of the form \( x^n \). According to this rule, the derivative of \( x^n \) with respect to \( x \) is \( n x^{n-1} \). It essentially "brings down" the exponent as a factor in front of the variable, and reduces the exponent by one.
This rule simplifies the process of differentiation. When applying the power rule, ensure all terms are in the form \( x^n \) before differentiating them. For example, in the exercise, the term \( rs^2 x^3 \) was processed by first recognizing it as a power of \( x \), and then applying the power rule to find its derivative as \( 3rs^2 x^2 \).
Overall, the power rule is a straightforward and effective method to quickly differentiate polynomial terms, making it a crucial tool for higher-level calculus problems.
This rule simplifies the process of differentiation. When applying the power rule, ensure all terms are in the form \( x^n \) before differentiating them. For example, in the exercise, the term \( rs^2 x^3 \) was processed by first recognizing it as a power of \( x \), and then applying the power rule to find its derivative as \( 3rs^2 x^2 \).
Overall, the power rule is a straightforward and effective method to quickly differentiate polynomial terms, making it a crucial tool for higher-level calculus problems.
Constant Multiple Rule
The constant multiple rule allows for effortless differentiation in expressions involving constants. This rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function itself.
For example, if you have a function \( c \cdot f(x) \), where \( c \) is a constant, then the derivative is simply \( c \cdot f'(x) \).
Applying this to the second term of the exercise, \(-rx\), we see that \(-r\) is a constant. Hence, the derivative here is \(-r \cdot (1) = -r\), where \( 1 \) is the derivative of \( x \) with respect to itself.
This rule is most helpful when dealing with large constants, as it provides a quick way to factor them out during differentiation.
For example, if you have a function \( c \cdot f(x) \), where \( c \) is a constant, then the derivative is simply \( c \cdot f'(x) \).
Applying this to the second term of the exercise, \(-rx\), we see that \(-r\) is a constant. Hence, the derivative here is \(-r \cdot (1) = -r\), where \( 1 \) is the derivative of \( x \) with respect to itself.
This rule is most helpful when dealing with large constants, as it provides a quick way to factor them out during differentiation.
Derivative of a Constant
When it comes to finding the derivative of a constant, the rule is simple: the derivative of any constant number is zero.
In the context of differentiation, a constant is any term that does not involve the variable with respect to which you're differentiating. This means that regardless of its value, a constant remains unaffected by changes in the variable.
For instance, in the expression \( s \) from the exercise, since \( s \) does not depend on \( x \), its derivative is \( 0 \). Recognizing constants and applying this rule significantly simplifies the differentiation process when such terms are present.
In the context of differentiation, a constant is any term that does not involve the variable with respect to which you're differentiating. This means that regardless of its value, a constant remains unaffected by changes in the variable.
For instance, in the expression \( s \) from the exercise, since \( s \) does not depend on \( x \), its derivative is \( 0 \). Recognizing constants and applying this rule significantly simplifies the differentiation process when such terms are present.
Polynomial Differentiation
Polynomial differentiation involves applying basic calculus rules, like the power and constant multiple rules, to functions that consist of terms where variables are raised to whole number exponents.
To differentiate a polynomial, follow these steps:
To differentiate a polynomial, follow these steps:
- Identify and differentiate each term individually.
- Apply the power rule to terms with the variable \( x \) raised to a power.
- Apply the constant multiple rule if a term is a constant times a variable.
- Use the derivative of a constant rule for terms without variables.
Other exercises in this chapter
Problem 29
Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \left(2 x^{3}-x\right) $$
View solution Problem 29
Differentiate $$ f(x)=a(x+1)(2 x-1) $$ with respect to \(x\). Assume that \(a\) is a positive constant.
View solution Problem 29
In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ y=2-|x
View solution Problem 29
Differentiate $$ f(x)=(a x+1)^{3} $$ with respect to \(x\). Assume that \(a\) is a positive constant.
View solution