Find dy/dx x^9y^8-y=x

Solution:

Step 1

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(x^9y^8 - y) = \frac{d}{dx}(x)$.

Step 2

Apply the derivative to the left-hand side of the equation.

Step 2.1

Utilize the Sum Rule to find the derivative of $x^9y^8 - y$: $\frac{d}{dx}[x^9y^8] + \frac{d}{dx}[-y]$.

Step 2.2

Find the derivative of $x^9y^8$ with respect to $x$.

Step 2.2.1

Use the Product Rule, which gives $\frac{d}{dx}[fg] = f\frac{d}{dx}[g] + g\frac{d}{dx}[f]$, where $f(x) = x^9$ and $g(x) = y^8$: $x^9\frac{d}{dx}[y^8] + y^8\frac{d}{dx}[x^9]$.

Step 2.2.2

Apply the Chain Rule to $\frac{d}{dx}[y^8]$.

Step 2.2.2.1

Set $u = y$ to apply the Chain Rule: $x^9\left(\frac{d}{du}[u^8]\frac{d}{dx}[y]\right) + y^8\frac{d}{dx}[x^9]$.

Step 2.2.2.2

Use the Power Rule, which states $\frac{d}{du}[u^n] = nu^{n-1}$ where $n = 8$: $x^9\left(8u^7\frac{d}{dx}[y]\right) + y^8\frac{d}{dx}[x^9]$.

Step 2.2.2.3

Replace $u$ with $y$: $x^9\left(8y^7\frac{d}{dx}[y]\right) + y^8\frac{d}{dx}[x^9]$.

Step 2.2.3

Express $\frac{d}{dx}[y]$ as $\frac{dy}{dx}$.

Step 2.2.4

Apply the Power Rule to $\frac{d}{dx}[x^9]$: $x^9(8y^7\frac{dy}{dx}) + y^8(9x^8)$.

Step 2.2.5

Rearrange the terms: $8x^9y^7\frac{dy}{dx} + 9y^8x^8$.

Step 2.2.6

Simplify the expression: $8x^9y^7\frac{dy}{dx} + 9y^8x^8$.

Step 2.3

Compute the derivative of $-y$ with respect to $x$.

Step 2.3.1

Since $-1$ is a constant, the derivative is $-\frac{d}{dx}[y]$.

Step 2.3.2

Express $\frac{d}{dx}[y]$ as $\frac{dy}{dx}$.

Step 2.4

Combine the terms: $8x^9y^7\frac{dy}{dx} + 9y^8x^8 - \frac{dy}{dx}$.

Step 3

Differentiate the right-hand side using the Power Rule: $\frac{d}{dx}[x] = 1$.

Step 4

Set the derivatives equal to each other: $8x^9y^7\frac{dy}{dx} + 9y^8x^8 - \frac{dy}{dx} = 1$.

Step 5

Isolate $\frac{dy}{dx}$.

Step 5.1

Subtract $9x^8y^8$ from both sides: $8x^9y^7\frac{dy}{dx} - \frac{dy}{dx} = 1 - 9x^8y^8$.

Step 5.2

Factor out $\frac{dy}{dx}$ from $8x^9y^7\frac{dy}{dx} - \frac{dy}{dx}$.

Step 5.3

Divide by the coefficient of $\frac{dy}{dx}$ and simplify.

Step 5.3.1

Divide each term by $8x^9y^7 - 1$.

Step 5.3.2

Simplify the equation to solve for $\frac{dy}{dx}$.

Step 5.3.3

Combine terms over a common denominator and simplify further if possible.

Step 6

Replace $y$ with $\frac{dy}{dx}$ to find the derivative: $\frac{dy}{dx} = \frac{1 - 9x^8y^8}{8x^9y^7 - 1}$.

Knowledge Notes:

1. Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.

2. Product Rule: For two functions $f(x)$ and $g(x)$, the derivative of their product is given by $\frac{d}{dx}[fg] = f\frac{d}{dx}[g] + g\frac{d}{dx}[f]$.

3. Chain Rule: If a function $y$ is a composite function of $u$ which is a function of $x$, then the derivative of $y$ with respect to $x$ is found by multiplying the derivative of $y$ with respect to $u$ by the derivative of $u$ with respect to $x$.

4. Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

5. Differentiation of a Constant: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

6. Solving for Derivatives: To isolate the derivative, one may need to use algebraic manipulation, such as factoring and simplifying expressions.