Determine $\frac{d W}{d T}$. Put Your Final Answer Completely In Terms Of $t$.Given:$w(x, Y, Z) = X^1 Y + Y^4 Z + Z^4$Where:$x(t) = 2t$, $y(t) = \sin(2.5t$\], $z(t) = \cos(2t$\]Calculate $\frac{d

Determine $\frac{d w}{d t}$

In this article, we will determine the derivative of a given function $w(x, y, z)$ with respect to time $t$. The function $w(x, y, z)$ is a composite function that depends on three variables $x$, $y$, and $z$, which are themselves functions of time $t$. We will use the chain rule to find the derivative of $w$ with respect to $t$.

The given function is:

$$w(x, y, z) = x^1 y + y^4 z + z^4$$

The functions of time $t$ are given as:

$$x(t) = 2t$$

$$y(t) = \sin(2.5t)$$

$$z(t) = \cos(2t)$$

$$$$

To find the derivative of $w$ with respect to $t$, we will use the chain rule. The chain rule states that if $w$ is a composite function of $x$, $y$, and $z$, which are themselves functions of $t$, then the derivative of $w$ with respect to $t$ is given by:

$$\frac{d w}{d t} = \frac{\partial w}{\partial x} \frac{d x}{d t} + \frac{\partial w}{\partial y} \frac{d y}{d t} + \frac{\partial w}{\partial z} \frac{d z}{d t}$$

We need to find the partial derivatives of $w$ with respect to $x$, $y$, and $z$.

$$\frac{\partial w}{\partial x} = y$$

$$\frac{\partial w}{\partial y} = x^1 + 4y^3 z$$

$$\frac{\partial w}{\partial z} = 4y^4$$

$$$$$$$$

We need to find the derivatives of $x$, $y$, and $z$ with respect to $t$.

$$\frac{d x}{d t} = 2$$

$$\frac{d y}{d t} = 2.5 \cos(2.5t)$$

$$\frac{d z}{d t} = -2 \sin(2t)$$

Now we can substitute the partial derivatives and derivatives into the chain rule formula.

$$\frac{d w}{d t} = y \cdot 2 + (x^1 + 4y^3 z) \cdot 2.5 \cos(2.5t) + 4y^4 \cdot (-2 \sin(2t))$$

We can simplify the expression by substituting the given functions of time $t$.

$$\frac{d w}{d t} = \sin(2.5t) \cdot 2 + (2t + 4\sin^3(2.5t) \cos(2.5t)) \cdot 2.5 \cos(2.5t) + 4\sin^4(2.5t) \cdot (-2 \sin(2t))$$

After simplifying the expression, we get:

$$\frac{d w}{d t} = 2\sin(2.5t) + 5t\cos(2.5t) + 10\sin^3(2.5t)\cos^2(2.5t) - 8\sin^5(2.5t)\sin(2t)$$

This is the final answer in terms of $t$.
Determine $\frac{d w}{d t}$: Q&A

In our previous article, we determined the derivative of a given function $w(x, y, z)$ with respect to time $t$. The function $w(x, y, z)$ is a composite function that depends on three variables $x$, $y$, and $z$, which are themselves functions of time $t$. We used the chain rule to find the derivative of $w$ with respect to $t$. In this article, we will answer some frequently asked questions related to the problem.

A: The chain rule is a formula for finding the derivative of a composite function. It states that if $w$ is a composite function of $x$, $y$, and $z$, which are themselves functions of $t$, then the derivative of $w$ with respect to $t$ is given by:

$$\frac{d w}{d t} = \frac{\partial w}{\partial x} \frac{d x}{d t} + \frac{\partial w}{\partial y} \frac{d y}{d t} + \frac{\partial w}{\partial z} \frac{d z}{d t}$$

$$$$$$$$

A: The partial derivatives of $w$ with respect to $x$, $y$, and $z$ are:

$$\frac{\partial w}{\partial x} = y$$

$$\frac{\partial w}{\partial y} = x^1 + 4y^3 z$$

$$\frac{\partial w}{\partial z} = 4y^4$$

$$$$$$$$

A: The derivatives of $x$, $y$, and $z$ with respect to $t$ are:

$$\frac{d x}{d t} = 2$$

$$\frac{d y}{d t} = 2.5 \cos(2.5t)$$

$$\frac{d z}{d t} = -2 \sin(2t)$$

A: To substitute the partial derivatives and derivatives into the chain rule formula, simply plug in the values of the partial derivatives and derivatives into the formula.

$$\frac{d w}{d t} = y \cdot 2 + (x^1 + 4y^3 z) \cdot 2.5 \cos(2.5t) + 4y^4 \cdot (-2 \sin(2t))$$

A: To simplify the expression, use algebraic manipulations to combine like terms and eliminate any unnecessary terms.

A: The final answer is:

$$\frac{d w}{d t} = 2\sin(2.5t) + 5t\cos(2.5t) + 10\sin^3(2.5t)\cos^2(2.5t) - 8\sin^5(2.5t)\sin(2t)$$

This is the final answer in terms of $t$.

In this article, we answered some frequently asked questions related to the problem of determining the derivative of a given function $w(x, y, z)$ with respect to time $t$. We used the chain rule to find the derivative of $w$ with respect to $t$ and simplified the expression to obtain the final answer.