To solve this problem, we will follow these steps:

Step 1: Differentiate the given equation implicitly

The given equation is:

$x^7y - xy^7 = -9999990$

Implicit differentiation will involve both $x$ and $y$, treating $y$ as a function of $x$. Taking the derivative of both sides with respect to $x$, we get:

$\frac{d}{dx}(x^7y) - \frac{d}{dx}(xy^7) = \frac{d}{dx}(-9999990)$

Using the product rule $(uv)' = u'v + uv'$:

$\frac{d}{dx}(x^7y) = 7x^6y + x^7\frac{dy}{dx}$ $\frac{d}{dx}(xy^7) = y^7 + 7xy^6\frac{dy}{dx}$

Substituting these into the equation:

$7x^6y + x^7\frac{dy}{dx} - (y^7 + 7xy^6\frac{dy}{dx}) = 0$

Simplify:

$7x^6y + x^7\frac{dy}{dx} - y^7 - 7xy^6\frac{dy}{dx} = 0$

Group terms involving $\frac{dy}{dx}$:

$x^7\frac{dy}{dx} - 7xy^6\frac{dy}{dx} = y^7 - 7x^6y$

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

$\frac{dy}{dx}(x^7 - 7xy^6) = y^7 - 7x^6y$

Solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{y^7 - 7x^6y}{x^7 - 7xy^6}$

Step 2: Evaluate $\frac{dy}{dx}$ at the point $(1, 10)$

Substitute $x = 1$ and $y = 10$ into the expression for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{10^7 - 7(1^6)(10)}{1^7 - 7(1)(10^6)}$

Simplify the numerator:

$10^7 - 7(1^6)(10) = 10000000 - 70 = 9999990$

Simplify the denominator:

$1^7 - 7(1)(10^6) = 1 - 7000000 = -6999999$

Thus:

$\frac{dy}{dx} = \frac{9999990}{-6999999} = -\frac{10}{7}$

Step 3: Write the equation of the tangent line

The slope of the tangent line is $-\frac{10}{7}$, and the point of tangency is $(1, 10)$. The equation of a line is:

$L(x) = y_1 + m(x - x_1)$

Substitute $y_1 = 10$, $m = -\frac{10}{7}$, and $x_1 = 1$:

$L(x) = 10 - \frac{10}{7}(x - 1)$

Simplify:

$L(x) = 10 - \frac{10}{7}x + \frac{10}{7}$

Combine constants:

$L(x) = \frac{70}{7} + \frac{10}{7} - \frac{10}{7}x = \frac{80}{7} - \frac{10}{7}x$

Thus, the equation of the tangent line is:

$L(x) = \frac{80}{7} - \frac{10}{7}x$

Do you have questions or need clarification? Here are some related questions to consider:

1. How do implicit differentiation and product rules work in other contexts?
2. What is the geometric interpretation of the tangent line at a point?
3. How do you verify the solution numerically or graphically?
4. Can this method extend to higher derivatives?
5. What happens to the solution when the constants in the equation are altered?

Tip: Double-check your substitutions and simplifications to avoid small errors in calculations!