Find \(\frac{dy}{dx}\)
Explanation
The relationship is simple: \(\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\). This suggests that to find the derivative of the inverse function, you take the derivative of that function with respect to \(y\), then reciprocate the derivative \(\left(\frac{1}{\frac{dx}{dy}}\right)\). It's the same thing as finding the derivative implicitly.
#1
\(x=y+y^2+y^3\)
\(\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(y+y^2+y^3\right)\)
The derivative of x is simple, it's 1. Now, distribute the d/dx for each term on the right
\(1=\frac{d}{dx}y+\frac{d}{dx}y^2+\frac{d}{dx}y^3\)
Perform the power rule
\(1=1\frac{dy}{dx}+2y\frac{dy}{dx}+3y^2\frac{d}{dx}\)
Factor the dy/dx on the right side
\(1=\frac{dy}{dx}\left(1+2y+3y^2\right)\)
Isolate for dy/dx
\(\frac{1}{1+2y+3y^2}=\frac{dy}{dx}\)
#2
\(2\left(4y+1\right)^3\)
Set the expression equal to x:
\(x=2\left(4y+1\right)^3\)
Setup the derivative:
\(\frac{d}{dx}x=\frac{d}{dx}\left[2\left(4y+1\right)^3\right]\)
Same as above, take the derivative with respect to x
\(1=\left[2\cdot 3\left(4y+1\right)^2\cdot 4\right]\frac{dy}{dx}\)
Derivative done, simplify:
\(\frac{1}{24\left(4y+1\right)^2}=\frac{dy}{dx}\)
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