How do I find partial derivatives with fractional exponents in the Chain Rule?

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In summary, we discussed finding partial derivatives for a given equation with a fractional exponent. We found that the product rule is used when the constant is a function of the variable being differentiated. We also saw that it is important to correctly apply the product rule and not include any unnecessary terms. Finally, we noted that partial derivatives are similar to ordinary derivatives and may seem easier at first, but eventually, they will become more challenging as the problems become more complex.
  • #1
harpazo
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Given k = ge^(x/y), find ∂k/∂x and ∂k/∂y. The fractional exponent throws me into a loop. Can someone show me step by step how to tackle this problem?
 
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  • #2
Harpazo said:
Given k = ge^(x/y), find ∂k/∂x and ∂k/∂y. The fractional exponent throws me into a loop. Can someone show me step by step how to tackle this problem?

If you can do $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x}\,\left( x\right) \end{align*}$ and $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}y}\,\left( \frac{1}{y} \right) \end{align*}$ there is no reason you can't do this problem.
 
  • #3
Prove It said:
If you can do $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x}\,\left( x\right) \end{align*}$ and $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}y}\,\left( \frac{1}{y} \right) \end{align*}$ there is no reason you can't do this problem.

d/dx [x] = 1

d/dy [1/y]

We know that 1/y can be expressed as y^(-1).

So, d/dy [y^(-1)] = -y^(-2), which of course is -1/y^2.
 
  • #4
Given k = g*e^(x/y), find ∂k/∂x.

∂k/∂x = g * e^(x/y) * (1/y) + e^(x/y)

∂k/∂x = [g*e^(x/y)]/(y) + e^(x/y)

Note: When taking the derivative of the exponent (x/y), I first expressed (x/y) as x*(1/y) and then took the partial derivative of x holding 1/y as constant. The product rule is also at work here. Is this correct?
 
Last edited:
  • #5
Given k = g*e^(x/y), find ∂k/∂y.

∂k/∂y = g * e^(x/y) * (-x/y^2) + e^(x/y)

I applied the product rule.
 
  • #6
Harpazo said:
Given k = g*e^(x/y), find ∂k/∂x.

∂k/∂x = g * e^(x/y) * (1/y) + e^(x/y)

∂k/∂x = [g*e^(x/y)]/(y) + e^(x/y)

Note: When taking the derivative of the exponent (x/y), I first expressed (x/y) as x*(1/y) and then took the partial derivative of x holding 1/y as constant. The product rule is also at work here. Is this correct?

If g is a constant then you don't need the product rule. If g is a function of x, then yes, the product rule will be needed. In either case, you have applied it incorrectly, as you should not have that extra "+ e^(x/y)"

- - - Updated - - -

Harpazo said:
Given k = g*e^(x/y), find ∂k/∂y.

∂k/∂y = g * e^(x/y) * (-x/y^2) + e^(x/y)

I applied the product rule.

Again, if g is a constant, you should not have used the product rule. If g is a function of y, then you would have needed the product rule, but you would have done so incorrectly, as again you should not have "+ e^(x/y)".
 
  • #7
Given k = g*e^(x/y), find ∂k/∂x.

∂k/∂x = g * e^(x/y) * (1/y)

∂k/∂x = [g*e^(x/y)]/(y)

Am I ok now?

Note: Here, y is held as constant.

- - - Updated - - -

Given k = g*e^(x/y), find ∂k/∂y.

∂k/∂y = g * e^(x/y) * (-x/y^2)

Am I ok now?

Note: I could also simplify further by placing y^2 in the denominator. Here, x is held as constant.
 
  • #8
Harpazo said:
Given k = g*e^(x/y), find ∂k/∂x.

∂k/∂x = g * e^(x/y) * (1/y)

∂k/∂x = [g*e^(x/y)]/(y)

Am I ok now?

Note: Here, y is held as constant.

- - - Updated - - -

Given k = g*e^(x/y), find ∂k/∂y.

∂k/∂y = g * e^(x/y) * (-x/y^2)

Am I ok now?

Note: I could also simplify further by placing y^2 in the denominator. Here, x is held as constant.

Yes they're both fine now :)
 
  • #9
Prove It said:
Yes they're both fine now :)

I find partial derivatives easier than calculus 1 derivatives.
 
  • #10
Harpazo said:
I find partial derivatives easier than calculus 1 derivatives.
The honeymoon isn't over yet. The partial derivatives act in the same way as the ordinary derivatives. It's just that your course hasn't given you the harder problems while you are learning new methods.

-Dan
 

FAQ: How do I find partial derivatives with fractional exponents in the Chain Rule?

What is the Chain Rule in Calculus 3?

The Chain Rule is a calculus method used to find the derivative of a composite function. It allows us to find the rate of change of a function that is composed of two or more functions.

How is the Chain Rule applied in Calculus 3?

The Chain Rule is applied by first identifying the inner and outer functions of the composite function. Then, we take the derivative of the outer function and multiply it by the derivative of the inner function. This gives us the derivative of the composite function.

Why is the Chain Rule important in Calculus 3?

The Chain Rule is important because it allows us to find the derivative of complicated composite functions that cannot be simplified using basic rules of differentiation. It is a fundamental concept in calculus and is used in various applications, such as optimization and related rates problems.

What are some common mistakes when using the Chain Rule in Calculus 3?

Some common mistakes when using the Chain Rule include not identifying the inner and outer functions correctly, forgetting to multiply by the derivative of the inner function, and making algebraic errors when simplifying the final derivative expression.

How can I practice and improve my understanding of the Chain Rule in Calculus 3?

To practice and improve your understanding of the Chain Rule, you can solve various problems involving composite functions and their derivatives. You can also seek help from online resources, such as practice quizzes and tutorials, or consult with a tutor or your instructor for additional support.

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