# Partial derivative chain rule for gradient

• physics2000
Thanks for correcting me. In summary, to compute the gradient of ln(z / (sqrt(x^2-y^2)), we can use the rules \log'(x)=\frac 1 x, (fg)'(x)=\frac{f'g-fg'}{g^2}, and \frac{d}{dx}x^a=ax^{a-1}. Alternatively, we can simplify the problem by using the hint e^(i Pi)+1=0.
physics2000

## Homework Statement

$$ln(z / (sqrt(x^2-y^2))$$

## Homework Equations

$$∇=(∂/(∂x)) + ... for y and z$$

I just want to know how to do the first term with respect to x

## The Attempt at a Solution

I am so rusty I don't know where to begin.

$ln(z) - \frac{1}{2}ln(x^2-y^2)$

That ought to make it easier. Now treat y and z as constants.

jfgobin said:
And you sum.

You don’t sum... Gradient is a vector and what you have found are the linearly independent components of it.

jfgobin, you shouldn't give away the final answer when you're answering posts in the homework forum. Just give a hint, like e^(i Pi)+1=0 did.
Mod note: I dealt with this.
physics2000, The hint given by e^(i Pi)+1=0 simplifies the problem significantly, but you don't have to use it. It's also possible to use these three rules directly:
\begin{align}
&\log'(x)=\frac 1 x\\
&(fg)'(x)=\frac{f'g-fg'}{g^2}\\
&\frac{d}{dx}x^a=ax^{a-1}
\end{align}

Last edited by a moderator:
cosmic dust said:
You don’t sum... Gradient is a vector and what you have found are the linearly independent components of it.

## 1. What is the partial derivative chain rule for gradient?

The partial derivative chain rule for gradient is a mathematical rule that describes how to calculate the gradient of a function that depends on multiple variables. It states that the gradient of a function with respect to one variable is equal to the partial derivative of that variable multiplied by the gradient of the function with respect to the other variables.

## 2. Why is the partial derivative chain rule important?

The partial derivative chain rule is important because it allows us to calculate the rate of change of a function with respect to multiple variables. This is useful in many fields of science, especially in physics and engineering, where many physical quantities depend on multiple variables.

## 3. How do you apply the partial derivative chain rule in practice?

To apply the partial derivative chain rule, you first need to identify the function that you want to take the gradient of. Then, you need to find the partial derivatives of that function with respect to each of its variables. Finally, you can use the chain rule formula to calculate the gradient of the function.

## 4. Can the partial derivative chain rule be used for any type of function?

Yes, the partial derivative chain rule can be used for any type of function, as long as the function depends on multiple variables. It is a general rule that applies to all differentiable functions.

## 5. How is the partial derivative chain rule related to other differentiation rules?

The partial derivative chain rule is closely related to other differentiation rules, such as the product rule and the quotient rule. In fact, these rules can be derived from the chain rule. Additionally, the chain rule can also be extended to higher dimensions, which is known as the multivariable chain rule.

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