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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?
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?
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.
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?
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.
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.
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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.
Prove It said:Yes they're both fine now :)
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.Harpazo said:I find partial derivatives easier than calculus 1 derivatives.
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.
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.
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.
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.
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.