Question

question: use implicit differentiation to find the equation of the tangent line to the function defined implicitly by the equation below at the point (2, - 2). (x^{6}+y^{4}=80). give your answer in the form (y = mx + b). provide your answer below: (y=square)

Explanation:

Step1: Differentiate implicitly

Differentiate $x^{6}+y^{4}=80$ with respect to $x$: $6x^{5}+4y^{3}y' = 0$.

Step2: Solve for $y'$

$y'=-\frac{6x^{5}}{4y^{3}}=-\frac{3x^{5}}{2y^{3}}$.

Step3: Find slope at $(2, - 2)$

Substitute $x = 2$ and $y=-2$ into $y'$: $m=-\frac{3\times2^{5}}{2\times(-2)^{3}}=6$.

Step4: Find $b$

Use $y=mx + b$ with $x = 2$, $y=-2$ and $m = 6$: $-2=6\times2 + b$, so $b=-14$.

Answer:

$y = 6x-14$