Problem

Given the equation below, find $\frac{d y}{d x}$.
\[
-33 x^{2}+4 x^{33} y+y^{7}=-28
\]
\[
\frac{d y}{d x}=
\]
Now, find the equation of the tangent line to the curve at $(1,1)$. Write your answer in $m x+b$ format
\[
y=
\]

Answer

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Answer

So, the equation of the tangent line to the curve at the point (1,1) is \(\boxed{y = -8x + 9}\).

Steps

Step 1 :First, we differentiate the given equation implicitly with respect to \(x\).

Step 2 :Using the chain rule, we get \(-66x + 4 \cdot 33x^{32}y + 4x^{32}y' + 7y^{6}y' = 0\).

Step 3 :Rearranging the terms, we get \(y' = \frac{66x - 132x^{32}y}{4x^{32} + 7y^{6}}\).

Step 4 :Substituting the point (1,1) into the equation, we get \(y' = \frac{66 - 132}{4 + 7} = -8\).

Step 5 :So, the slope of the tangent line at the point (1,1) is -8.

Step 6 :Using the point-slope form of a line, we get the equation of the tangent line as \(y - 1 = -8(x - 1)\).

Step 7 :Simplifying the equation, we get \(y = -8x + 9\).

Step 8 :So, the equation of the tangent line to the curve at the point (1,1) is \(\boxed{y = -8x + 9}\).

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