# Partial derivative with chain rule: check work

• 939
In summary, the problem involves evaluating y at t=0 using the chain rule, given the equations y = 10kl - √k - √l, k = (t/5) + 5, and l = 5e^t/10. The solution involves using the partial derivatives of y with respect to k and l, multiplied by the derivatives of k and l with respect to t. The result is approximately 51, with k and l having values of 5 at t=0.
939

## Homework Statement

If possible, please check my work for any large errors.

y = 10kl - √k - √l
k = (t/5) + 5
l = 5e^t/10

Evaluate at t = 0 using chain rule.

## Homework Equations

y = 10kl - √k - √l
k = (t/5) + 5
l = 5e^t/10

## The Attempt at a Solution

= ∂y/∂k * dk/dt + ∂y/∂l * dl/dt
= (10l - 0.5K^-0.5)(1/5) + (10k - 0.5l^-.05)((e^(x/10))/2)

k(0) = 5
l(0) = 5
y(0) = ~9.95 + ~41.03 = ~51

939 said:

## Homework Statement

If possible, please check my work for any large errors.

y = 10kl - √k - √l
k = (t/5) + 5
l = 5e^t/10

Evaluate at t = 0 using chain rule.

## Homework Equations

y = 10kl - √k - √l
k = (t/5) + 5
l = 5e^t/10

## The Attempt at a Solution

= ∂y/∂k * dk/dt + ∂y/∂l * dl/dt
= (10l - 0.5K^-0.5)(1/5) + (10k - 0.5l^-.05)((e^(x/10))/2)
The above is close. The -0.5l^(-.05) term should be -0.5l^(-0.5). I don't know if that was a transcription error. Also, the exponential term would be clearer as (1/2)e(x/10).
939 said:
k(0) = 5
l(0) = 5
y(0) = ~9.95 + ~41.03 = ~51
k(0) and l(0) are fine, but I didn't check your other numbers.

One other thing. It's not good to start your first line with "=". For your problem, you should start with dy/dt = ...

1 person

## What is the chain rule and how does it apply to partial derivatives?

The chain rule is a calculus rule that allows us to find the derivative of a composite function. In the context of partial derivatives, it helps us find the rate of change of a multivariable function with respect to one of its variables while holding the other variables constant.

## How do I know if I have applied the chain rule correctly in a partial derivative?

To check if you have correctly applied the chain rule in a partial derivative, you should make sure that the inner and outer functions are properly identified and that the derivative of the inner function is multiplied by the derivative of the outer function.

## What are some common mistakes when using the chain rule in partial derivatives?

Some common mistakes when using the chain rule in partial derivatives include forgetting to identify the inner and outer functions, misapplying the power rule, and not properly simplifying the expression after applying the chain rule.

## Why is it important to check our work when using the chain rule in partial derivatives?

Checking our work when using the chain rule in partial derivatives is important to ensure that we have correctly applied the rule and have not made any algebraic or computational errors. This helps us avoid incorrect solutions and better understand the concepts involved.

## Are there any tips for effectively using the chain rule in partial derivatives?

Some tips for effectively using the chain rule in partial derivatives include carefully identifying the inner and outer functions, practicing with different types of functions, and simplifying the expression as much as possible before taking the derivative.

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