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Solution assignment 07 Implicit differentiation

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Assignment 7

Calculate y'' if:

xy=5

Solution

Implicit differentiation can also used to calculate the second derivative. In this case the product rule in combination with the quotient rule is applied:

y'=-\displaystyle\frac{y}{x}

We can differentiate again:

y''=-\displaystyle\frac{xy'-y}{x^2}=-\displaystyle\frac{-2y}{x^2}

We can verify the result because the original expression can easily be written in explicit form:

y=\displaystyle\frac{5}{x}

and then the first and second derivative can be found in the usual way. Verify this.

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