Exercise · ZW-IMPLICIT-DIFFERENTIATION-009 · ★★☆ Intermediate
If \(xy^{2}=8\), what is \(\dfrac{dy}{dx}\)?
Choose one option.
Show answer
- A. \(- \frac{y}{2x}\)
Explanation
By the product rule: \(y^{2}+2xy\,y'=0\), so \(y'=-\frac{y}{2x}\).
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.
\[x^{2}+y^{2}=25\ \Rightarrow\ 2x+2y\,y'=0\ \Rightarrow\ y'=-\frac{x}{y}\]
More practice problems: implicit differentiation
- ZW-IMPLICIT-DIFFERENTIATION-001 If x^2+y^2=25, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-002 If xy=1, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-003 If x^3+y^3=6xy, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-004 If sin y=x, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-005 If x^2+xy+y^2=7, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-006 If x^2-y^2=4, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-007 If y^2=2x, what is (dy)/(dx)?
- ZW-IMPLICIT-DIFFERENTIATION-008 If e^y=x, what is (dy)/(dx)?
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