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Find df/dx and df/dy for f(x, y) = (3x^5 y^4 + 5)^2.

Question:

Find {eq}\frac{\mathrm{d}f}{\mathrm{d}x} {/eq} and {eq}\frac{\mathrm{d}f}{\mathrm{d}y} {/eq} for {eq}f(x, y) = (3x^5 y^4 + 5)^2 {/eq}.

Answer and Explanation:

To find the derivative we will proceed as

{eq}f(x,y)=(3x^{5}y^{4}+5)^{2} {/eq}

Differentiating it with respect to x using the chain rule

{eq}f=u^{n}\\ f'=nu^{n-1}u' {/eq}

Let us use the rule and find the derivative now{eq}\frac{\mathrm{d} f}{\mathrm{d} x}=2\left ( 3x^{5}y^{4}+5 \right )15x^{4}y^{4} {/eq}

Now differentiating the function with respect to y and treating x as constant

{eq}\frac{\mathrm{d} f}{\mathrm{d} y}=2\left ( 3x^{5}y^{4}+5 \right )3x^{5}4y^{3} {/eq}


Learn more about this topic:

The Chain Rule for Partial Derivatives

from GRE Math: Study Guide & Test Prep

Chapter 14 / Lesson 4
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