Derivative of e^(-(1/3)x^2-(1/5)y^4) ?

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In summary, the derivative of e^(-(1/3)x^2-(1/5)y^4) is {-(2/3)x-[((4/5)y^3)dy/dx]}e^(-(1/3)x^2-(1/5)y^4).
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Derivative of e^(-(1/3)x^2-(1/5)y^4) ?

I need help figuring this out. I know that the derivative of e^5x is 5e^x but I'm not sure what to do when x is raised to a power or when there is another variable involved, such as y. Once again here is what I need to find: Derivative of e^(-(1/3)x^2-(1/5)y^4)

And if you guys know of a way that I can display that in a clearer way such as with some type of symbol tool please let me know for future posts. To make that clearer I want to find the derivative of e raised to the power of (negative one-third x^2 minus one-fifth y^4). Also, if you know of a site that fully explains the derivatives of e and includes explanations to the above problem let me know. Thanks.
 
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It may just be a typo on your part, but the derivative of e^5x is 5e^5x, not 5e^x.

With derivatives of e^(stuff), you start out by simply re-writing e^(stuff) and then multiplying by the derivative of (stuff). Like in e^5x, you re-write e^5x and then multiply by the derivative of 5x, which is five, giving you 5e^5x.

In your problem, you re-write your original term and then multiply by the derivative of
(-(1/3)x^2-(1/5)y^4) which is (-(2/3)x-[((4/5)y^3)dy/dx]) using implicit differentiation. The final answer would then be {-(2/3)x-[((4/5)y^3)dy/dx]}e^(-(1/3)x^2-(1/5)y^4).
 
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Related to Derivative of e^(-(1/3)x^2-(1/5)y^4) ?

1. What is the formula for the derivative of e^(-(1/3)x^2-(1/5)y^4)?

The formula for the derivative of e^(-(1/3)x^2-(1/5)y^4) is e^(-(1/3)x^2-(1/5)y^4) * (-2/3)x - (4/5)y^3.

2. How do you find the derivative of e^(-(1/3)x^2-(1/5)y^4)?

To find the derivative of e^(-(1/3)x^2-(1/5)y^4), you need to use the chain rule and the power rule. First, you need to take the derivative of the exponent, which is -(1/3)x^2-(1/5)y^4. Then, you multiply it by the derivative of the inside function, which is e^(-(1/3)x^2-(1/5)y^4). Finally, you multiply it by the derivative of the outside function, which is -2/3x - 4/5y^3.

3. What is the significance of the negative exponents in the derivative of e^(-(1/3)x^2-(1/5)y^4)?

The negative exponents in the derivative of e^(-(1/3)x^2-(1/5)y^4) indicate that the function is decreasing as x and y increase. This can also be seen in the graph of the function, which has a downward slope.

4. How does the derivative of e^(-(1/3)x^2-(1/5)y^4) relate to the original function?

The derivative of e^(-(1/3)x^2-(1/5)y^4) is a function that represents the rate of change of the original function. It tells us how fast the original function is changing at a specific point. The derivative can also be used to find the slope of the tangent line at any point on the graph of the original function.

5. Can the derivative of e^(-(1/3)x^2-(1/5)y^4) be simplified?

Yes, the derivative of e^(-(1/3)x^2-(1/5)y^4) can be simplified to (-2/3)x * e^(-(1/3)x^2-(1/5)y^4) - (4/5)y^3 * e^(-(1/3)x^2-(1/5)y^4), using the power rule and the chain rule. However, it is usually not necessary to simplify the derivative unless explicitly stated in the problem.

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