Problem 11
Question
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$\frac{\partial}{\partial m}\left(\frac{1}{2} m v^{2}\right)$$
Step-by-Step Solution
Verified Answer
The partial derivative is \( \frac{1}{2} v^2 \).
1Step 1: Identify the Function
The function given is \( \frac{1}{2} mv^2 \). This expression represents the kinetic energy of an object in terms of its mass \( m \) and velocity \( v \).
2Step 2: Recognize the Variable of Differentiation
We need to differentiate the function with respect to the variable \( m \). Therefore, \( m \) is the variable, and \( v \) is considered a constant in this context.
3Step 3: Apply the Power Rule
To find the partial derivative, apply the power rule for differentiation. The function can be rewritten as \( \frac{1}{2} m (v^2) \), where \( v^2 \) is a constant with respect to \( m \). The derivative of \( m \) is just \( 1 \).
4Step 4: Calculate the Partial Derivative
Differentiate \( \frac{1}{2} m (v^2) \) with respect to \( m \). Since \( v^2 \) is constant, the partial derivative becomes \( \frac{1}{2} \, (v^2) \, \cdot \, \frac{d}{dm}(m) = \frac{1}{2} v^2 \).
Key Concepts
DifferentiationKinetic EnergyPower Rule
Differentiation
Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function changes. In simpler terms, it is about computing the 'slope' or 'derivative' of a function at any given point. To differentiate a function, one needs to understand the behavior and changes of the variables involved.
When we consider partial differentiation, which is used in multivariable calculus, we focus on how a function changes with respect to one variable while keeping the others constant. This is crucial when dealing with functions of more than one variable, like in physics or engineering. For instance, if you have a function related to kinetic energy that depends on mass and velocity, and you wish to know how the energy varies with just the mass, you would find the partial derivative with respect to the mass.
When we consider partial differentiation, which is used in multivariable calculus, we focus on how a function changes with respect to one variable while keeping the others constant. This is crucial when dealing with functions of more than one variable, like in physics or engineering. For instance, if you have a function related to kinetic energy that depends on mass and velocity, and you wish to know how the energy varies with just the mass, you would find the partial derivative with respect to the mass.
- Partial derivatives are noted with the symbol \( \frac{\partial}{\partial x} \).
- It is important to remember which variable you are differentiating with respect to, as others remain constant in this process.
Kinetic Energy
Kinetic energy is the energy of motion. It describes how much energy an object possesses due to its motion. The formula for kinetic energy is given by \( \frac{1}{2} mv^2 \), where \( m \) stands for mass, and \( v \) represents velocity.
Both mass and velocity play integral roles in determining the amount of kinetic energy an object has. Velocity has a squared term, showing that increases in speed have a much larger effect on the energy compared to mass. The relationship between kinetic energy and these variables can be deeply analyzed by employing calculus, specifically partial differentiation, to see how changes in one variable affect the energy while keeping the other constant.
Both mass and velocity play integral roles in determining the amount of kinetic energy an object has. Velocity has a squared term, showing that increases in speed have a much larger effect on the energy compared to mass. The relationship between kinetic energy and these variables can be deeply analyzed by employing calculus, specifically partial differentiation, to see how changes in one variable affect the energy while keeping the other constant.
- If you increase the mass while keeping velocity constant, kinetic energy increases linearly.
- Increasing velocity while keeping the mass constant leads to a quadratic change in energy.
Power Rule
The power rule is a basic but essential tool in calculus used for finding the derivative of functions expressed as powers of a variable. According to the power rule, if you have a function \( x^n \), where \( n \) is a constant, the derivative of this function with respect to \( x \) is \( nx^{n-1} \).
Applying the power rule is straightforward when dealing with particular derivatives, as we saw with the kinetic energy function. When differentiating the expression \( \frac{1}{2} mv^2 \) with respect to \( m \), you treat \( v^2 \) as a constant. Thus, the expression simplifies using the power rule, yielding a partial derivative of \( \frac{1}{2}v^2 \).
Applying the power rule is straightforward when dealing with particular derivatives, as we saw with the kinetic energy function. When differentiating the expression \( \frac{1}{2} mv^2 \) with respect to \( m \), you treat \( v^2 \) as a constant. Thus, the expression simplifies using the power rule, yielding a partial derivative of \( \frac{1}{2}v^2 \).
- The power rule is instantaneous and efficient, making it a go-to method for differentiation.
- The rule applies to any differentiable function where the variable is raised to a power.
Other exercises in this chapter
Problem 11
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