Use implicit differentiation to find the slope of the tangent line to the curve.

Question

Use implicit differentiation to find the slope of the tangent line to the curve

y/(x+8y) = x^6 - 4

at the point (1, -3/25)

what is the slope?

Answer 1

Step 1: Differentiate implicitly

Starting with the equation:

\[

\frac{y}{x + 8y} = x^6 - 4

\]

You differentiate both sides with respect to \(x\). For the left side, you need to use the quotient rule:

\[

\frac{d}{dx}\left(\frac{y}{x + 8y}\right) = \frac{(x + 8y)\frac{dy}{dx} - y\left(1 + 8\frac{dy}{dx}\right)}{(x + 8y)^2}

\]

For the right side, you need to use power rule and constant rule:

\[

\frac{d}{dx}(x^6 - 4) = 6x^5

\]

Setting the two derivatives equal:

\[

\frac{(x + 8y)\frac{dy}{dx} - y\left(1 + 8\frac{dy}{dx}\right)}{(x + 8y)^2} = 6x^5

\]

Step 2: Plug in the point \((1, -\frac{3}{25})\)

Substituting \(x = 1\) and \(y = -\frac{3}{25}\) into the equation:

\[

\frac{(1 + 8(-\frac{3}{25}))\frac{dy}{dx} - \left(-\frac{3}{25}\right)\left(1 + 8\frac{dy}{dx}\right)}{(1 + 8(-\frac{3}{25}))^2} = 6(1)^5

\]

Simplifying each part:

\(1 + 8(-\frac{3}{25}) = 1 - \frac{24}{25} = \frac{1}{25}\)

So the denominator becomes \(\left(\frac{1}{25}\right)^2 = \frac{1}{625}\)

Now, substitute these into the equation:

\[

\frac{\frac{1}{25}\frac{dy}{dx} + \frac{3}{25}(1 + 8\frac{dy}{dx})}{\frac{1}{625}} = 6

\]

Multiply through by \(625\) to clear the fraction in the denominator:

\[

625\left(\frac{1}{25}\frac{dy}{dx} + \frac{3}{25}(1 + 8\frac{dy}{dx})\right) = 6

\]

Simplify inside the parentheses:

\[

25\left(\frac{dy}{dx} + 3(1 + 8\frac{dy}{dx})\right) = 6

\]

Distribute:

\[

25\frac{dy}{dx} + 75 + 600\frac{dy}{dx} = 6

\]

Combine like terms:

\[

625\frac{dy}{dx} = 6 - 75

\]

\[

625\frac{dy}{dx} = -69

\]

So,

\[

\frac{dy}{dx} = \frac{-69}{625}=-0.1104

\]

So, the slope of the tangent line to the curve at the given point is -0.1104.

Please note that the above text is given in LaTeX format. Please convert it to Word or PDF format to see equations more clearly.

Answer 2

y / (x+8y) = x⁶ - 4

Using the quotient rule, [(x+8y)(dy/dx) - y(1+8dy/dx)] / (x+8y)² = 6x⁵

At the point (1, -3/25), we have [(1/25)(dy/dx) + (3/25)(1+8(dy/dx))] / (1/25)² = 6

25(dy/dx) + 75(1+8(dy/dx)) = 6

625(dy/dx) = -69

dy/dx = -69/625 = -0.1104