- #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.
MHB Don't solving this problem, just need some info.
- Thread starter Nate Learning
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Starting with the equation \((3x^7 + y^2)^{1/2} = \sin^2(y) + 100xy\), squaring both sides is an option but may not simplify the differentiation process. Instead, differentiating directly using the chain rule is recommended. The derivatives of both sides are calculated, leading to an equation involving \(\frac{dy}{dx}\). The final step is to solve this equation for \(\frac{dy}{dx}\). This approach provides a clearer path to finding the derivative without complicating the initial equation.
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 $\frac{dy}{dx}$.
Hello!
I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem.
Given:
##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0##
##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1##
##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0##
I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...