Problem 20 - Advanced Engineering Mathematics

Question

Find the first partial derivatives of the given function. $$ z=\left(-x^{4}+7 y^{2}+3 y\right)^{6} $$

Step-by-Step Solution

Answer

The partial derivatives are \( \frac{\partial z}{\partial x} = 6(-x^4 + 7y^2 + 3y)^5(-4x^3) \) and \( \frac{\partial z}{\partial y} = 6(-x^4 + 7y^2 + 3y)^5(14y + 3) \).

Step 1: Recognize the Function Form

The function given is of the form \( z = u^6 \), where \( u = -x^4 + 7y^2 + 3y \). We need to use the chain rule to find the partial derivatives with respect to \( x \) and \( y \).

Step 2: Differentiate with Respect to x

First, apply the chain rule for the derivative with respect to \( x \):\[ \frac{\partial z}{\partial x} = 6u^5 \frac{\partial u}{\partial x}. \]Find \( \frac{\partial u}{\partial x} \):\[ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(-x^4 + 7y^2 + 3y) = -4x^3. \]Thus, the first partial derivative of \( z \) with respect to \( x \) is:\[ \frac{\partial z}{\partial x} = 6(-x^4 + 7y^2 + 3y)^5(-4x^3). \]

Step 3: Differentiate with Respect to y

Apply the chain rule for the derivative with respect to \( y \):\[ \frac{\partial z}{\partial y} = 6u^5 \frac{\partial u}{\partial y}. \]Find \( \frac{\partial u}{\partial y} \):\[ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(-x^4 + 7y^2 + 3y) = 14y + 3. \]Thus, the first partial derivative of \( z \) with respect to \( y \) is:\[ \frac{\partial z}{\partial y} = 6(-x^4 + 7y^2 + 3y)^5(14y + 3). \]

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental technique used to differentiate composite functions. Essentially, it helps in finding the derivative of a function composed of two or more functions.
The chain rule states that if you have a composite function \( z = f(g(x)) \), then the derivative of this function can be obtained by multiplying the derivative of the outer function evaluated at the inner function times the derivative of the inner function:

• \( \frac{dz}{dx} = f'(g(x)) \cdot g'(x) \)

In the given exercise, function \( z = (-x^4 + 7y^2 + 3y)^6 \)is composed such that \( u = -x^4 + 7y^2 + 3y \), and \( z = u^6 \).
When applying the chain rule here, we first differentiate the outer function followed by differentiating the inner function, just like following the chain links. This rule is especially useful when dealing with functions that have nested compositions, simplifying the process into more manageable steps.

First Partial Derivative

Partial derivatives are used when dealing with functions of multiple variables to determine how the function changes with respect to one of those variables while keeping others constant.
In the context of our exercise, we focus on finding the first partial derivatives of \( z \) with respect to \( x \) and \( y \).
Partial derivatives allow us to understand the influence of each variable on the function separately:

• \( \frac{\partial z}{\partial x} \) represents how \( z \) changes as \( x \) changes, with \( y \) held constant.
• \( \frac{\partial z}{\partial y} \) indicates how \( z \) changes as \( y \) changes, with \( x \) held constant.

In the solution, using the chain rule, we first calculated

• \( \frac{\partial u}{\partial x} = -4x^3 \)
• \( \frac{\partial u}{\partial y} = 14y + 3 \)

Then, these partial derivatives of \( u \) are used to find the partial derivatives of \( z \).

Function Differentiation

Differentiation of a function is a mathematical process to find the derivative, which represents the rate of change of the function's value with respect to changes in its input values. It helps understand how a function behaves, and provides important insights into the role of each variable.
For functions of multiple variables like \( z = (-x^4 + 7y^2 + 3y)^6 \),differentiation involves taking partial derivatives to analyze the effect of changing each variable:

• For the function with respect to \( x \), we found \( \frac{\partial z}{\partial x} = 6(-x^4 + 7y^2 + 3y)^5(-4x^3) \).


• For the function with respect to \( y \), we derived \( \frac{\partial z}{\partial y} = 6(-x^4 + 7y^2 + 3y)^5(14y + 3) \).

These operations show how \( z \) varies as \( x \) and \( y \) change, offering valuable perspectives in optimizing and analyzing functions in multiple dimensions. Differentiation serves as the mathematical foundation for further studies in calculus and applied mathematics.