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I am trying to understand the difference between a partial derivative and a total derivative when one component is defined as a function of another

If I have a function such as $f(x,y(x)) = x+y$

and $y(x) = x^2$

Is it true that the total derivative with respect to $x$ is $1 + 2x$

While the partial with respect to $x$ is $1$

Thanks

Caligula
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F. Slyin
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    I believe this post answers your question: http://math.stackexchange.com/questions/174270/what-exactly-is-the-difference-between-a-derivative-and-a-total-derivative – MathematicianByMistake Mar 27 '16 at 15:11

1 Answers1

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$f(x,y)=x+y$

To find out the partial derivative with respect to $x$, we treat $y$ as a constant, and differentiate the whole expression w.r.t. $x$.

$$\frac{\partial f}{\partial x}=\frac{dx}{dx}=1$$

Total derivative is given by:

$$df=\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy$$

So $$\frac{df}{dx}=1+ 1*\frac{dy}{dx}=1+2x$$

lEm
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