Nate Learning
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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.
Taking the derivative of an equation with a radical
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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.
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skeeter
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Looking at the square of the right side, i probably would not.
Opalg
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MHB
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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:
jonah1
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Beer induced query follows.
From which book did you get this challenging derivative?Nate Learning said:
HOI
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$(3x^7+ y^2)^{1/2}= sin^2(y)+ 100xy$
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 $\frac{dy}{dx}$.