If e^y=x, what is (dy)/(dx)?

Exercise · ZW-IMPLICIT-DIFFERENTIATION-008 · ★★☆ Intermediate

If \(e^{y}=x\), what is \(\dfrac{dy}{dx}\)?

Choose one option.

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B. \(\frac{1}{x}\)

Explanation

Differentiating gives \(e^{y}y'=1\), and since \(e^{y}=x\), we get \(y'=\frac{1}{x}\).

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Review the concept: Implicit differentiation

When \(y\) is defined implicitly by an equation in \(x\) and \(y\), both sides are differentiated with respect to \(x\), remembering that \(y\) depends on \(x\) (chain rule), and then \(\frac{dy}{dx}\) is solved for.

\[x^{2}+y^{2}=25\ \Rightarrow\ 2x+2y\,y'=0\ \Rightarrow\ y'=-\frac{x}{y}\]

If \(e^{y}=x\), what is \(\dfrac{dy}{dx}\)?

Explanation

Review the concept: Implicit differentiation

More practice problems: implicit differentiation