Find dy/dx x^9y^7-y=x

Solution:

Step:1

Apply the derivative operator to both sides of the given equation: $\frac{d}{dx}(x^9y^7 - y) = \frac{d}{dx}(x)$.

Step:2

Take the derivative of the left-hand side of the equation.

Step:2.1

Utilize the Sum Rule to separate the derivative: $\frac{d}{dx}(x^9y^7) + \frac{d}{dx}(-y)$.

Step:2.2

Compute $\frac{d}{dx}(x^9y^7)$.

Step:2.2.1

Apply the Product Rule, which states $\frac{d}{dx}(fg) = f\frac{dg}{dx} + g\frac{df}{dx}$ where $f(x) = x^9$ and $g(x) = y^7$: $x^9\frac{d}{dx}(y^7) + y^7\frac{d}{dx}(x^9)$.

Step:2.2.2

Use the Chain Rule for $\frac{d}{dx}(y^7)$, which implies $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$ where $f(u) = u^7$ and $g(x) = y$.

Step:2.2.2.1

Set $u = y$ to apply the Chain Rule: $x^9(7u^6\frac{dy}{dx}) + y^7\frac{d}{dx}(x^9)$.

Step:2.2.2.2

Employ the Power Rule, which states $\frac{d}{du}(u^n) = nu^{n-1}$ where $n = 7$: $x^9(7y^6\frac{dy}{dx}) + y^7\frac{d}{dx}(x^9)$.

Step:2.2.2.3

Substitute $u$ back with $y$: $x^9(7y^6\frac{dy}{dx}) + y^7\frac{d}{dx}(x^9)$.

Step:2.2.3

Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$.

Step:2.2.4

Apply the Power Rule to $\frac{d}{dx}(x^9)$, yielding $9x^8$.

Step:2.2.5

Rearrange to place the constant 7 in front of $x^9$: $7x^9y^6\frac{dy}{dx} + y^7(9x^8)$.

Step:2.2.6

Place the constant 9 in front of $y^7$: $7x^9y^6\frac{dy}{dx} + 9y^7x^8$.

Step:2.3

Calculate $\frac{d}{dx}(-y)$.

Step:2.3.1

Since $-1$ is a constant, the derivative becomes $-\frac{dy}{dx}$: $7x^9y^6\frac{dy}{dx} + 9y^7x^8 - \frac{dy}{dx}$.

Step:2.3.2

Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$.

Step:2.4

Combine all terms: $7x^9y^6\frac{dy}{dx} + 9y^7x^8 - \frac{dy}{dx}$.

Step:3

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

Step:4

Set the derivatives of the left-hand side equal to the derivative of the right-hand side: $7x^9y^6\frac{dy}{dx} + 9y^7x^8 - \frac{dy}{dx} = 1$.

Step:5

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

Step:5.1

Subtract $9x^8y^7$ from both sides: $7x^9y^6\frac{dy}{dx} - \frac{dy}{dx} = 1 - 9x^8y^7$.

Step:5.2

Factor out $\frac{dy}{dx}$ from $7x^9y^6\frac{dy}{dx} - \frac{dy}{dx}$.

Step:5.2.1

Factor $\frac{dy}{dx}$ out of $7x^9y^6\frac{dy}{dx}$: $\frac{dy}{dx}(7x^9y^6) - \frac{dy}{dx}$.

Step:5.2.2

Factor $\frac{dy}{dx}$ out of $-\frac{dy}{dx}$: $\frac{dy}{dx}(7x^9y^6 - 1)$.

Step:5.2.3

Combine the factored terms: $\frac{dy}{dx}(7x^9y^6 - 1) = 1 - 9x^8y^7$.

Step:5.3

Divide both sides by $(7x^9y^6 - 1)$ to solve for $\frac{dy}{dx}$.

Step:5.3.1

Divide the equation by $(7x^9y^6 - 1)$: $\frac{\frac{dy}{dx}(7x^9y^6 - 1)}{7x^9y^6 - 1} = \frac{1 - 9x^8y^7}{7x^9y^6 - 1}$.

Step:5.3.2

Simplify the left side by canceling out the common factors.

Step:5.3.2.1

Cancel the common factor $(7x^9y^6 - 1)$: $\frac{dy}{dx} = \frac{1 - 9x^8y^7}{7x^9y^6 - 1}$.

Step:5.3.3

Simplify the right side by combining over the common denominator.

Step:6

Replace $y$ with $\frac{dy}{dx}$ to obtain the final derivative: $\frac{dy}{dx} = \frac{1 - 9x^8y^7}{7x^9y^6 - 1}$.

Knowledge Notes:

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

2. Product Rule: The derivative of a product of two functions is given by $d(uv)/dx = u(dv/dx) + v(du/dx)$.

3. Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

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

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

6. Implicit Differentiation: When a function is not given in the form of y explicitly defined by x, differentiation is performed with respect to x, treating y as a function of x when necessary. This often involves the application of the Chain Rule.