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Find the slope of the tangent line to the graph of $x = 5y^2 - 15$ at $(5,2)$ .

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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 = 5y^2 - 15

at

(5,2)

.

Find Derivative with Respect to $y$ : First, we need to find the derivative of $x$ with respect to $y$ , since the equation is implicitly defined. We differentiate both sides of the equation with respect to $y$ . $\frac{dx}{dy} = 10y$
Evaluate Derivative at $y=2$ : Next, we evaluate the derivative at the point $y = 2$ to find the slope of the tangent line at that specific point. $\newline$ $\frac{dx}{dy} = 10 \times 2 = 20$
Convert to $\frac{dy}{dx}$ : However, we need the slope of the tangent in terms of $\frac{dy}{dx}$ , not $\frac{dx}{dy}$ . Since $\frac{dy}{dx}$ is the reciprocal of $\frac{dx}{dy}$ , we calculate: $\newline$ $\frac{dy}{dx} = \frac{1}{\left(\frac{dx}{dy}\right)} = \frac{1}{20}$

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