Implicit Equations.

[ffuh205]

In finding the derivative for the implicit equation, y^2 + x^2 y^2 + x^4 = 5. Select the correct answers below. There may be multiple correct answers.

A. The derivative is (-4x^3 - 2xy^2)/(2y + 2yx^2).

B. In finding the derivative the first rule used would be the sum rule.

C. This equation has a vertical tangent line at x = 0.

D. In finding the derivative one would use the chain rule, and exponential rule as well as others.

E. In finding the derivative one would use the product rule, constant multiplier rule, and power rule as well as others.

F. In finding the derivative one MUST use the product rule.

G. This function is differentiable everywhere from negative infinity to positive infinity.

I have D & E.

 

[soroban]

Hello, ffuh205!

Find the derivative for the implicit equation: \(\displaystyle y^2 + x^2 y^2 + x^4 \:=\: 5\)
Select the correct answers below. There may be multiple correct answers.

(A) The derivative is (-4x^3 - 2xy^2)/(2y + 2yx^2).

This is true . . . but the answer is written really stupidly!


\(\displaystyle 2yy' +2x^2yy' + 2xy^2 + 4x^3 \:=\:0 \quad\Rightarrow\quad2yy' + 2x^2yy' \;=\;-4x^3-2xy^2\)

. . \(\displaystyle 2y(1 + x^2)y' \;=\;-2x(2x^2 + y^2) \quad\Rightarrow\quad y' \;=\;\frac{-2x(2x^2+y^2)}{2y(1+x^2)}\)

\(\displaystyle \text{Therefore: }\;y' \;=\;-\frac{x(2x^2+y^2)}{y(1+x^2)}\)

(B) In finding the derivative the first rule used would be the sum rule.

FALSE . . . The first rule is the Chain Rule.

(C) This equation has a vertical tangent line at x = 0.

FALSE

\(\displaystyle \text{If } x = 0.\text{ then: }\,y^2\,=\,5 \quad\Rightarrow\quad y \:=\:\pm\sqrt{5}\)

\(\displaystyle \text{Then: }\;y' \;=\;-\frac{0(0+5)}{\pm\sqrt{5}(1+0)} \;=\;0\)


\(\displaystyle \text{There is a }horizontal\text{ tangent at }x = 0.\)

(D) In finding the derivative one would use the chain rule, and exponential rule as well as others.

TRUE

(E) In finding the derivative one would use the product rule, constant multiplier rule, and power rule as well as others.

FALSE
The constant multiplier rule is not used.

(F) In finding the derivative one MUST use the product rule.

FALSE

\(\displaystyle \text{The function can be written: }\;y^2 \;=\;\frac{5-x^4}{1+x^2}\)

. . and the Product Rule is not used.

(G) This function is differentiable everywhere from negative infinity to positive infinity.

FALSE

\(\displaystyle \text{The derivative is: }\;y' \;=\;-\frac{x(2x^2+y^2)}{y(1+x^2)}\)


\(\displaystyle \text{The derivative is undefined when }y \,=\,0\)

\(\displaystyle \text{If }y = 0\text{, then: }x^4 \:=\:5 \quad\Rightarrow\quad x \:=\:\pm\sqrt[4]{5}\)

\(\displaystyle \text{There are vertical asymptotes at: }\,x \:=\:\pm\sqrt[4]{5}\)

 

[ffuh205]

Thanks. Your thoughts on the other similar questions I posted?