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Find the implicit derivative $\frac{d}{dx}\left(x^xy=\sqrt[3]{\frac{x\left(x+1\right)\left(x-2\right)}{\left(x^2+1\right)\left(2x+3\right)}}\right)$

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Solution

Formulas

Videos

Implicit differentiation | Advanced derivatives | AP Calculus AB | Khan Academy

https://www.youtube.com/watch?v=mSVrqKZDRF4

Calculus - Take the log of both sides to find the derivative, y = (x(x^2 + 1)^2)/(sqrt(2x^2 - 1))

https://www.youtube.com/watch?v=7aF6Ck6ZRxw

Worked example: Evaluating derivative with implicit differentiation | AP Calculus AB | Khan Academy

https://www.youtube.com/watch?v=KyYC8XzKsHU

Algebra 2 - How to find the real zero of a cubic function, y = -1(x - 3)^3 + 1

https://www.youtube.com/watch?v=4dSA02Fzric

Tutorial How to find the difference between two rational expressions, ((x-2)/(x^2 -8x+12))-(3/(x-2))

https://www.youtube.com/watch?v=IqvMZFULWdo

Calculus: Derivatives 2 | Taking derivatives | Differential Calculus | Khan Academy

https://www.youtube.com/watch?v=ay8838UZ4nM

Function Plot

Plotting: $\left(\ln\left(x\right)+1\right)x^xy+x^xy^{\prime}=\frac{\left(\left(x+1\right)\left(x-2\right)+x\left(x-2+x+1\right)\right)\left(x^2+1\right)\left(2x+3\right)+\left(-x-1\right)x\left(x-2\right)\left(2x\left(2x+3\right)+2\left(x^2+1\right)\right)}{3\left(x^2+1\right)^2\left(2x+3\right)^2}\sqrt[3]{\left(\frac{\left(x^2+1\right)\left(2x+3\right)}{x\left(x+1\right)\left(x-2\right)}\right)^{2}}$

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log
log◻
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d/dx
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sin
cos
tan
cot
sec
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asin
acos
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sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

Used Formulas

See formulas (7)

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