- #1
Nate Learning
- 2
- 0
Should I start off by squaring both sides to get rid of the radical on the left? and then start the derivative process? Thank you.
Don't solving this problem, just need some info.
- Context: MHB
- Thread starter Nate Learning
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Discussion Overview
The discussion revolves around the process of differentiating an equation involving a radical expression. Participants explore different approaches to tackle the derivative, specifically whether to square both sides of the equation or differentiate directly as it stands. The context is primarily mathematical reasoning related to calculus.
Discussion Character
- Mathematical reasoning
Main Points Raised
- Some participants propose starting by squaring both sides to eliminate the radical before differentiating.
- Others argue that differentiating the equation as it is may be just as effective, suggesting the use of the chain rule directly on the original equation.
- A participant provides a detailed breakdown of the derivatives involved, including the derivatives of both sides of the equation and the resulting expressions.
- There is a question regarding the source of the problem, indicating a possible interest in the context or difficulty of the derivative challenge.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the best approach to take for differentiating the equation, with multiple competing views on whether to square both sides or differentiate directly.
Contextual Notes
Some assumptions about the ease or difficulty of each method are not explicitly stated, and the discussion does not resolve which method is ultimately preferable.
Should I start off by squaring both sides to get rid of the radical on the left? and then start the derivative process? Thank you.
- #2
skeeter
- 1,103
- 1
Looking at the square of the right side, i probably would not.
- #3
Opalg
Gold Member
MHB
- 2,778
- 13
You certainly could do it that way, but I don't see that it would be any easier than just differentiating the equation as it stands. Write it as $$\bigl(3x^7 + y^2\bigr)^{1/2} = \sin^2y + 100xy,$$ and differentiate both sides with respect to $x$ using the chain rule.Nate Learning said:
- #4
jonah1
- 107
- 0
Beer induced query follows.
From which book did you get this challenging derivative?Nate Learning said:
- #5
HOI
- 921
- 2
$(3x^7+ y^2)^{1/2}= sin^2(y)+ 100xy$
The derivative of $3x^7+ y^2$ with respect to x is $21x^6+ 2y\frac{dy}{dx}$ so the derivative of $(3x^7+ y^2)^{1/2}$ is $\frac{1}{2}(3x^7+ y^2)^{-1/2}(21x^6+ 2y\frac{dy}{dx})$.
The derivative of $sin^2(y)$ with respect to x is $2 sin(y) cos(y)\frac{dy}{dx}$.
And the derivative of $100 xy$ with respect to x is $100y+ 100x\frac{dy}{dx}$.
So the derivative of $(3x^7+y^2)^{1/2}= sin^2(y)+ 100xy$ with respect to x is $\frac{1}{2}(3x^7+ y^2)^{-1/2}(21x^6+ 2y\frac{dy}{dx})= 2 sin(y) cos(y)\frac{dy}{dx}+ 100y+ 100x\frac{dy}{dx}$.
Solve that equation for $\frac{dy}{dx}$.
The derivative of $3x^7+ y^2$ with respect to x is $21x^6+ 2y\frac{dy}{dx}$ so the derivative of $(3x^7+ y^2)^{1/2}$ is $\frac{1}{2}(3x^7+ y^2)^{-1/2}(21x^6+ 2y\frac{dy}{dx})$.
The derivative of $sin^2(y)$ with respect to x is $2 sin(y) cos(y)\frac{dy}{dx}$.
And the derivative of $100 xy$ with respect to x is $100y+ 100x\frac{dy}{dx}$.
So the derivative of $(3x^7+y^2)^{1/2}= sin^2(y)+ 100xy$ with respect to x is $\frac{1}{2}(3x^7+ y^2)^{-1/2}(21x^6+ 2y\frac{dy}{dx})= 2 sin(y) cos(y)\frac{dy}{dx}+ 100y+ 100x\frac{dy}{dx}$.
Solve that equation for $\frac{dy}{dx}$.
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