Finding Derivative Question

[kellemax000]

Hi,
When I'm given an equation and told to find the derivative I'm good at using the quotient rule, product rule, power rule, chain rule, constant rule, constant multiple rule, and I can implicitly differentiate. But when I have to differentiate implicitly and during the process I have to use the other rules, I use them get an answer but then when I change the equation slightly so that you can differentiate using a simpler rule the answers never match up. How would you go about implicitly differentiating #8?

 

[DrPhil]

Hi,
When I'm given an equation and told to find the derivative I'm good at using the quotient rule, product rule, power rule, chain rule, constant rule, constant multiple rule, and I can implicitly differentiate. But when I have to differentiate implicitly and during the process I have to use the other rules, I use them get an answer but then when I change the equation slightly so that you can differentiate using a simpler rule the answers never match up. How would you go about implicitly differentiating #8?

Very hard to read, but I am guessing...\(\displaystyle \sqrt{xy} = x - 2y\)

The right side is easy, but the left requires power, chain, and product rules.

\(\displaystyle \dfrac{d}{dx}\sqrt{xy} = \dfrac{1}{2\ \sqrt{xy}} \times \dfrac{d}{dx}(xy) =
\dfrac{1}{2\ \sqrt{xy}} \times (x\ \dfrac{dy}{dx} + y) \)

 

[HallsofIvy]

Very hard to read, but I am guessing...\(\displaystyle \sqrt{xy} = x - 2y\)

The right side is easy, but the left requires power, chain, and product rules.

\(\displaystyle \dfrac{d}{dx}\sqrt{xy} = \dfrac{1}{2\ \sqrt{xy}} \times \dfrac{d}{dx}(xy) =
\dfrac{1}{2\ \sqrt{xy}} \times (x\ \dfrac{dy}{dx} + y) \)

Another way of doing the same thing- write \(\displaystyle \sqrt{xy}= x^{1/2}y^{1/2}\).
So the derivative with respect to x is, using the product rule, \(\displaystyle (x^{1/2})'y^{1/2}+ x^{1/2}(y^{1/2})'\)
And using the "power rule" that is \(\displaystyle (1/2)x^{-1/2}y^{1/2}+ (1/2)x^{1/2}y^{-1/2}(dy/dx)\).