How do you use implicit differentiation to find the slope of the curve given 12(x^2+y^2)=25xy at (3,4)?

Jul 30, 2017

Explanation:

Given: $12 \left({x}^{2} + {y}^{2}\right) = 25 x y$

Use the distributive property:

$12 {x}^{2} + 12 {y}^{2} = 25 x y$

Differentiate each term:

$\frac{d}{\mathrm{dx}} 12 {x}^{2} + \frac{d}{\mathrm{dx}} 12 {y}^{2} = \frac{d}{\mathrm{dx}} 25 x y$

The first term is just the power rule:

$24 x + \frac{d}{\mathrm{dx}} 12 {y}^{2} = \frac{d}{\mathrm{dx}} 25 x y$

The second term requires the chain rule:

$24 x + 24 y \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} 25 x y$

The last term requires the use of the product rule:

$24 x + 24 y \frac{\mathrm{dy}}{\mathrm{dx}} = 25 y + 25 x \frac{\mathrm{dy}}{\mathrm{dx}}$

Collect all of the terms containing $\frac{\mathrm{dy}}{\mathrm{dx}}$ on the left and the other terms on the right:

$24 y \frac{\mathrm{dy}}{\mathrm{dx}} - 25 x \frac{\mathrm{dy}}{\mathrm{dx}} = 25 y - 24 x$

Factor out $\frac{\mathrm{dy}}{\mathrm{dx}}$:

$\left(24 y - 25 x\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 25 y - 24 x$

Divide both sides by the leading factor on the left:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{25 y - 24 x}{24 y - 25 x}$

The slope, m, of the tangent line is the above evaluated at $\left(3 , 4\right)$:

$m = \frac{25 \left(4\right) - 24 \left(3\right)}{24 \left(4\right) - 25 \left(3\right)}$

$m = \frac{4}{3}$