Find dy/dx x^8+y^7=6

Solution:

Step 1

Apply differentiation to both sides of the equation with respect to $x$: $\frac{d}{dx}(x^8+y^7)=\frac{d}{dx}(6)$.

Step 2

Differentiate the left-hand side term by term.

Step 2.1

Perform differentiation.

Step 2.1.1

Utilize the Sum Rule in differentiation: the derivative of a sum is the sum of the derivatives. Thus, $\frac{d}{dx}(x^8)+\frac{d}{dx}(y^7)$.

Step 2.1.2

Apply the Power Rule, which says the derivative of $x^n$ is $n{x}^{n-1}$, to $x^8$ to get $8x^7$. For $y^7$, since $y$ is a function of $x$, keep it as $\frac{d}{dx}(y^7)$: $8x^7+\frac{d}{dx}(y^7)$.

Step 2.2

Determine $\frac{d}{dx}(y^7)$.

Step 2.2.1

Use the Chain Rule for differentiation: the derivative of $f(g(x))$ is $f^{\prime }(g(x))g^{\prime }(x)$. Here, let $f(x)=x^7$ and $g(x)=y$.

Step 2.2.1.1

Introduce $u=y$ to apply the Chain Rule: $8x^7+\frac{d{u}^7}{du}\cdot \frac{dy}{dx}$.

Step 2.2.1.2

Differentiate $u^7$ using the Power Rule to get $7u^6$: $8x^7+7u^6\cdot \frac{dy}{dx}$.

Step 2.2.1.3

Substitute $u$ back with $y$: $8x^7+7y^6\cdot \frac{dy}{dx}$.

Step 2.2.2

Express $\frac{dy}{dx}$ as $y^{\prime }$: $8x^7+7y^6{y}^{\prime }$.

Step 3

The derivative of a constant, such as 6, with respect to $x$ is 0.

Step 4

Combine the differentiated terms: $8x^7+7y^6{y}^{\prime }=0$.

Step 5

Isolate $y^{\prime }$ (which is $\frac{dy}{dx}$).

Step 5.1

Subtract $8x^7$ from both sides: $7y^6{y}^{\prime }=-8x^7$.

Step 5.2

Divide by $7y^6$ to solve for $y^{\prime }$.

Step 5.2.1

Divide each term by $7y^6$: $\frac{7y^6{y}^{\prime }}{7y^6}=\frac{-8x^7}{7y^6}$.

Step 5.2.2

Simplify the left side.

Step 5.2.2.1

Cancel the common factor of 7: $\frac{y^6{y}^{\prime }}{y^6}=\frac{-8x^7}{7y^6}$.

Step 5.2.2.2

Cancel the common factor of $y^6$: $y^{\prime }=\frac{-8x^7}{7y^6}$.

Step 5.2.3

Simplify the right side: $y^{\prime }=-\frac{8x^7}{7y^6}$.

Step 6

Replace $y^{\prime }$ with $\frac{dy}{dx}$: $\frac{dy}{dx}=-\frac{8x^7}{7y^6}$.

Knowledge Notes:

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

2. Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $n{x}^{n-1}$.

3. Chain Rule: If a variable $y$ depends on $u$ which in turn depends on $x$, then the derivative of $y$ with respect to $x$ is the product of the derivative of $y$ with respect to $u$ and the derivative of $u$ with respect to $x$.

4. Differentiation of Constants: The derivative of a constant with respect to any variable is zero.

5. Implicit Differentiation: When a function $y$ is defined implicitly by an equation in $x$ and $y$, we differentiate both sides of the equation with respect to $x$ and then solve for $\frac{dy}{dx}$.

6. Simplifying Fractions: When dividing by a fraction, we can simplify by canceling out common factors in the numerator and denominator.