Problem 5
Question
Differentiate the given function by applying the theorems of this section. $$ F(t)=\frac{1}{4} t^{4}-\frac{1}{2} t^{2} $$
Step-by-Step Solution
Verified Answer
The derivative of \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \) is \( F'(t) = t^3 - t \).
1Step 1: Identify the function components
The function given is \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \). This function is composed of two terms: \( \frac{1}{4} t^4 \) and \( -\frac{1}{2} t^2 \).
2Step 2: Differentiate each term separately
To differentiate the function, apply the power rule \( \frac{d}{dt} [t^n] = n t^{n-1} \). First, differentiate the term \( \frac{1}{4} t^4 \). The derivative is \( \frac{1}{4} \times 4 t^{4-1} = t^3 \). Next, differentiate the term \( -\frac{1}{2} t^2 \). The derivative is \( -\frac{1}{2} \times 2 t^{2-1} = -t \).
3Step 3: Combine the derivatives
Now, combine the two derivatives obtained in the previous step: \( t^3 \) and \( -t \). Therefore, the derivative of the function is \( F'(t) = t^3 - t \).
Key Concepts
Power RuleDerivativeFunction Components
Power Rule
When differentiating functions, the power rule is a useful and straightforward tool. It states that if you have a term in the form of \( t^n \), the derivative of this term is \( n t^{n-1} \). For example, if you need to differentiate \( t^4 \), you use the power rule:
\[ \frac{d}{dt} [t^4] = 4 t^{4-1} = 4 t^3 \] The power rule makes differentiation easy because it only requires a basic operation of multiplying and decreasing the exponent by one.
This rule applies generally to terms where the variable is raised to a power. For instance, in the original function \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \), each term is a power of \( t \), making the power rule an ideal method to differentiate them.
\[ \frac{d}{dt} [t^4] = 4 t^{4-1} = 4 t^3 \] The power rule makes differentiation easy because it only requires a basic operation of multiplying and decreasing the exponent by one.
This rule applies generally to terms where the variable is raised to a power. For instance, in the original function \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \), each term is a power of \( t \), making the power rule an ideal method to differentiate them.
Derivative
The derivative of a function gives us the rate at which the function is changing at any given point. It's like saying how steep a hill is at any particular spot.
To find the derivative of a function, you apply differentiation rules such as the power rule. The derivative tells us how the function's output value changes with respect to changes in the input value. For instance, in our function \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \), the derivative \( F'(t) \) highlights how \( F(t) \) changes as \( t \) changes.
Derivatives have widespread applications, including finding maximum and minimum values of functions, optimizing areas such as velocity in physics, and even in complex areas like machine learning.
To find the derivative of a function, you apply differentiation rules such as the power rule. The derivative tells us how the function's output value changes with respect to changes in the input value. For instance, in our function \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \), the derivative \( F'(t) \) highlights how \( F(t) \) changes as \( t \) changes.
Derivatives have widespread applications, including finding maximum and minimum values of functions, optimizing areas such as velocity in physics, and even in complex areas like machine learning.
Function Components
A function may consist of various components or terms comprising constants, variables, and coefficients. Understanding these components is crucial in the differentiation process.
In the function provided, \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \), we have two main components:
Each term is differentiated separately, ensuring the power rule is applied correctly. The first term, \( \frac{1}{4} t^4 \), is separated into its coefficient \( \frac{1}{4} \) and the variable term \( t^4 \). The second term, \( -\frac{1}{2} t^2 \), similarly breaks down into \( -\frac{1}{2} \) and \( t^2 \).
By handling each component individually, we can solve complex functions step by step, making differentiation more manageable. Combining these individually differentiated components gives us the final derivative of the function.
In the function provided, \( F(t) = \frac{1}{4} t^4 - \frac{1}{2} t^2 \), we have two main components:
- \( \frac{1}{4} t^4 \)
- \( -\frac{1}{2} t^2 \)
Each term is differentiated separately, ensuring the power rule is applied correctly. The first term, \( \frac{1}{4} t^4 \), is separated into its coefficient \( \frac{1}{4} \) and the variable term \( t^4 \). The second term, \( -\frac{1}{2} t^2 \), similarly breaks down into \( -\frac{1}{2} \) and \( t^2 \).
By handling each component individually, we can solve complex functions step by step, making differentiation more manageable. Combining these individually differentiated components gives us the final derivative of the function.
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