Math 22 Differentials and Marginal Analysis

Differentials

 Let $y=f(x)$ represent a differentiable function. The differential of $x$ (denoted by $dx$)
 is any nonzero real number. The differential of $y$ (denoted by ) is $dy=f'(x)dx$.

Example: 1) Consider the function $f(x)=3x^{3}$. Find $dy$ when $x=1$ and $dx=0.01$

Solution:
Notice: $f(x)=3x^{3}$, so $dy=f'(x)dx=9x^{2}dx=9(1)^{2}.(0.01)=0.09$

2) Find $dy$ of each function below:

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

Solution:
Notice: $y={\frac {5x+7}{15}}={\frac {5x}{15}}+{\frac {7}{15}}$, so
$dy=f'(x)dx={\frac {5}{15}}={\frac {1}{3}}dx$

b) $y=x(1.25+0.02{\sqrt {x}})$

Solution:
Notice: $y=x(1.25+0.02{\sqrt {x}})=1.25x+0.02x{\sqrt {x}}=1.25x+0.02x^{\frac {3}{2}}$, so $dy=f'(x)dx=1.25dx+0.02({\frac {3}{2}})x^{\frac {1}{2}}={\frac {1}{3}}dx$

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