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Find the slope of the tangent line to the graph of $x = 3y^3 - 4$ at $(-1,1)$ .

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Find tangent lines using implicit differentiation

Full solution

Q. Find the slope of the tangent line to the graph of

x = 3y^3 - 4

at

(-1,1)

.

Find Derivative with Respect to $y$ : First, we need to find the derivative of $x$ with respect to $y$ to get the slope of the tangent line. Since $x = 3y^3 - 4$ , differentiate both sides with respect to $y$ . $\newline$ $\frac{dx}{dy} = 9y^2$
Substitute $y = 1$ : Next, substitute $y = 1$ into the derivative to find the slope at the point $(-1,1)$ . $\newline$ Slope = $9*(1)^2 = 9$
Convert Slope to $dy/dx$ : However, we need the slope of the tangent in terms of $dx/dy$ , but the slope of the tangent line to the curve at any point is given by $dy/dx$ , which is the reciprocal of $dx/dy$ . Slope of the tangent line = $1/(9)$

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