nplicit differentiation to determine $$ \frac{d y}{d x} $$ for the equation $$ x+y^{5}=7 $$

Question

UpStudy AI · Accepted Answer

To find $$ \frac{dy}{dx} $$ using implicit differentiation for the equation $$ x + y^5 = 7 $$, we will follow these steps:

1. Differentiate both sides of the equation with respect to $$ x $$.
2. Apply the chain rule where necessary.
3. Solve for $$ \frac{dy}{dx} $$.

Let's start with the equation:

$$
x + y^5 = 7
$$

### Step 1: Differentiate both sides

Differentiating the left side:

- The derivative of $$ x $$ with respect to $$ x $$ is $$ 1 $$.
- For $$ y^5 $$, we use the chain rule: the derivative is $$ 5y^4 \frac{dy}{dx} $$.

So, differentiating the left side gives:

$$
1 + 5y^4 \frac{dy}{dx}
$$

Differentiating the right side:

- The derivative of $$ 7 $$ is $$ 0 $$.

Putting it all together, we have:

$$
1 + 5y^4 \frac{dy}{dx} = 0
$$

### Step 2: Solve for $$ \frac{dy}{dx} $$

Now, we can isolate $$ \frac{dy}{dx} $$:

$$
5y^4 \frac{dy}{dx} = -1
$$

Dividing both sides by $$ 5y^4 $$:

$$
\frac{dy}{dx} = -\frac{1}{5y^4}
$$

### Final Result

Thus, the derivative $$ \frac{dy}{dx} $$ is:

$$
\frac{dy}{dx} = -\frac{1}{5y^4}
$$
$$ \frac{dy}{dx} = -\frac{1}{5y^4} $$

nplicit differentiation to determine \( \frac{d y}{d x} \) for the equation \( x+y^{5}=7 \)