Problem 21
Question
Find the derivative of the function. \(f(x)=4 x^{4}-3 x^{5 / 2}+2\)
Step-by-Step Solution
Verified Answer
The derivative of the function \(f(x) = 4x^4 - 3x^{5/2} + 2\) is: \(f'(x) = 16x^3 - \frac{15}{2}x^{1/2}\).
1Step 1: Identify the Terms
First, identify the terms in the function:
1. \(4x^4\)
2. \(-3x^{5/2}\)
3. \(2\)
2Step 2: Apply the Power Rule to the First Term
For the first term, \(4x^4\), apply the power rule:
\(\frac{d}{dx}(4x^4) = 4\cdot(4x^{4-1}) = 16x^3\)
3Step 3: Apply the Power Rule to the Second Term
For the second term, \(-3x^{5/2}\), apply the power rule:
\(\frac{d}{dx}(-3x^{5/2}) = -3\cdot(\frac{5}{2}x^{(5/2)-1}) = -\frac{15}{2}x^{1/2}\)
4Step 4: Apply the Power Rule to the Third Term
For the third term, \(2\), apply the power rule:
\(\frac{d}{dx}(2) = 0\)
Since the derivative of a constant is zero.
5Step 5: Combine the Results
Combine the derivatives of each term to find the overall derivative of the function:
\(\frac{d}{dx}(4x^4 - 3x^{5/2} + 2) = 16x^3 - \frac{15}{2}x^{1/2} + 0\)
The final answer is:
\(f'(x) = 16x^3 - \frac{15}{2}x^{1/2}\)
Key Concepts
Power RuleCalculating DerivativesDerivative of a PolynomialConstant Term Differentiation
Power Rule
Calculating the derivative of a function involving exponents is a fundamental skill in calculus, and the power rule is a straightforward method for doing this. In general, if you have a function of the form
\( f(x) = ax^n \), where \( a \) is a constant and \( n \) is a real number, then the derivative of \( f(x) \) with respect to \( x \) is given by
\( f'(x) = anx^{n-1} \).
This rule is applied by taking the exponent \( n \), multiplying the entire term by \( n \), and then decreasing the exponent by 1.
For example, using the power rule on the term \( 4x^4 \) in our exercise, we multiply the constant 4 by the exponent 4 and then subtract one from the exponent to get
\( 16x^3 \) as the derivative of that term.
\( f(x) = ax^n \), where \( a \) is a constant and \( n \) is a real number, then the derivative of \( f(x) \) with respect to \( x \) is given by
\( f'(x) = anx^{n-1} \).
This rule is applied by taking the exponent \( n \), multiplying the entire term by \( n \), and then decreasing the exponent by 1.
For example, using the power rule on the term \( 4x^4 \) in our exercise, we multiply the constant 4 by the exponent 4 and then subtract one from the exponent to get
\( 16x^3 \) as the derivative of that term.
Calculating Derivatives
Derivatives measure how a function's output changes as its input changes. The process of calculating derivatives is called differentiation. When differentiating a function like \( f(x) = 4x^4 - 3x^{5/2} + 2 \), we treat each term separately, applying rules of differentiation for each type of term.
The power rule, as described previously, handles terms with exponents. It is one of several techniques you can use. For more complex functions, you might need product rules, quotient rules, or chain rules. But in cases like our exercise, the power rule and the understanding that the derivative of a constant is zero are sufficient for finding the derivative of the entire expression.
The power rule, as described previously, handles terms with exponents. It is one of several techniques you can use. For more complex functions, you might need product rules, quotient rules, or chain rules. But in cases like our exercise, the power rule and the understanding that the derivative of a constant is zero are sufficient for finding the derivative of the entire expression.
Derivative of a Polynomial
Polynomials are algebraic expressions made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The derivative of a polynomial is found by applying the power rule to each term individually and then combining the results.
In our exercise, we differentiate a polynomial function, \( f(x) = 4x^4 - 3x^{5/2} + 2 \). Each term is tackled separately: first \( 4x^4 \), then \( 3x^{5/2} \), and finally the constant term. These derivatives are then summed to give the overall derivative of the polynomial. By systematically applying the power rule, we ensure that no term is overlooked and that the final derivative represents the rate of change of the entire polynomial function.
In our exercise, we differentiate a polynomial function, \( f(x) = 4x^4 - 3x^{5/2} + 2 \). Each term is tackled separately: first \( 4x^4 \), then \( 3x^{5/2} \), and finally the constant term. These derivatives are then summed to give the overall derivative of the polynomial. By systematically applying the power rule, we ensure that no term is overlooked and that the final derivative represents the rate of change of the entire polynomial function.
Constant Term Differentiation
One of the simpler rules in calculus is that the derivative of a constant term is zero. This is because constants do not change, and so the rate at which they change with respect to any variable is zero.
In the context of differentiating the polynomial function in our exercise, the constant term \( 2 \) has a derivative of zero. Although it may seem trivial, remembering to include constant terms in the differentiation process is important. They often act as placeholders that ensure the structure of the function remains intact, even though they do not contribute to the function's rate of change.
In the context of differentiating the polynomial function in our exercise, the constant term \( 2 \) has a derivative of zero. Although it may seem trivial, remembering to include constant terms in the differentiation process is important. They often act as placeholders that ensure the structure of the function remains intact, even though they do not contribute to the function's rate of change.
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