Math 22 Differentials and Marginal Analysis

Differentials

 Let $y=f(x)$ represent a differentiable function. The differential of $x$ (denoted by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dx}
)
 is any nonzero real number. The differential of $y$ (denoted by ) is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy=f'(x)dx}
.

Example: 1) Consider the function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=3x^{3}} . Find $dy$ when $x=1$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dx=0.01}

Solution:
Notice: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=3x^{3}} , so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy=f'(x)dx=9x^{2}dx=9(1)^{2}.(0.01)=0.09}

2) Find $dy$ of each function below:

a) $y={\frac {5x+7}{15}}$

Solution:
Notice: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y={\frac {5x+7}{15}}={\frac {5x}{15}}+{\frac {7}{15}}} , so
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy=f'(x)dx={\frac {5}{15}}={\frac {1}{3}}dx}

b) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=x(1.25+0.02{\sqrt {x}})}

Solution:

Notice: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=x(1.25+0.02{\sqrt {x}})=1.25x+0.02x{\sqrt {x}}=1.25x+0.02x^{\frac {3}{2}}} , so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy=f'(x)dx=[1.25+0.02({\frac {3}{2}})x^{\frac {1}{2}}]dx=[1.25+0.03{\sqrt {x}}]dx}

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