Chapter 4: Q5P (page 201)

If we are given $z = z (x, y)$ and $y = y (x)$ , show that the chain rule (5.1) gives $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$ $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$

Short Answer

Expert verified

The chain rule $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$ is proved.

Step by step solution

Explanation of solution

The provided expressions are $y = y (x)$ and $z = z (x, y)$ .

Chain Rule

The number of functions in the composition affects how many differentiation steps are required when using the chain rule to get the derivative of composite functions.

Calculation

Consider the relation $y = y (x)$

Here, z can be expressed as,

$z = z (x, y (x)) = z (x)$

Therefore, $\frac{d z}{d x}$ exists.

Apply the chain rule.

$\frac{d z}{d x} = \frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t} - - - (1)$

Substitute in equation (1).

$\frac{d z}{d x} = \frac{\partial z}{\partial x} \frac{d x}{d x} + \frac{\partial z}{\partial y} \frac{d y}{d x} = \frac{\partial z}{\partial x} (1) + \frac{\partial z}{\partial y} \frac{d y}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$

Hence the chain rule $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$ is proved.

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If we are given $z = z (x, y)$ and $y = y (x)$ , show that the chain rule (5.1) gives $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$ $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

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If we are given z=z(x,y)and y=y(x), show that the chain rule (5.1) givesdzdx=∂z∂x+∂z∂ydydxdzdx=∂z∂x+∂z∂ydydx

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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If we are given $z = z (x, y)$ and $y = y (x)$ , show that the chain rule (5.1) gives $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$ $\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}$