Question

Find {d y}{d x} if x(8ʸ )=4 x⁵ +3 y⁴ {d y}/{d x}=

Answer 1

To find \(\frac{{dy}}{{dx}}\) when \(x \cdot 8^y = 4x^5 + 3y^4\), we'll use implicit differentiation.

Given the equation:
\[x \cdot 8^y = 4x^5 + 3y^4\]

Let's differentiate both sides of the equation with respect to \(x\):

For the left-hand side \(x \cdot 8^y\):
- Apply the product rule: \(x \cdot 8^y\)
- The derivative of \(x\) with respect to \(x\) is \(1\).
- The derivative of \(8^y\) with respect to \(x\) is \(8^y \cdot \ln(8) \cdot \frac{{dy}}{{dx}}\) using the chain rule.

For the right-hand side \(4x^5 + 3y^4\):
- Differentiate each term separately.
- The derivative of \(4x^5\) with respect to \(x\) is \(20x^4\) using the power rule.
- The derivative of \(3y^4\) with respect to \(x\) is \(12y^3 \cdot \frac{{dy}}{{dx}}\) using the chain rule.

Setting up the equation:
\[1 \cdot 8^y + x \cdot \ln(8) \cdot 8^y \cdot \frac{{dy}}{{dx}} = 20x^4 + 12y^3 \cdot \frac{{dy}}{{dx}}\]

Now, let's solve for \(\frac{{dy}}{{dx}}\):
\[8^y + x \cdot \ln(8) \cdot 8^y \cdot \frac{{dy}}{{dx}} = 20x^4 + 12y^3 \cdot \frac{{dy}}{{dx}}\]

Group the terms involving \(\frac{{dy}}{{dx}}\) on one side:
\[x \cdot \ln(8) \cdot 8^y \cdot \frac{{dy}}{{dx}} - 12y^3 \cdot \frac{{dy}}{{dx}} = 20x^4 - 8^y\]

Factor out \(\frac{{dy}}{{dx}}\):
\[\frac{{dy}}{{dx}} \cdot (x \cdot \ln(8) \cdot 8^y - 12y^3) = 20x^4 - 8^y\]

Finally, solve for \(\frac{{dy}}{{dx}}\):
\[\frac{{dy}}{{dx}} = \frac{{20x^4 - 8^y}}{{x \cdot \ln(8) \cdot 8^y - 12y^3}}\]

This expression gives the value of \(\frac{{dy}}{{dx}}\) in terms of the given variables \(x\) and \(y\).

Answer 2

The question requires finding the derivative of y with respect to x given an equation involving x and y. Implicit differentiation is used, applying the product rule and then differentiating each part separately. The exact derivative will depend on the method applied and additional context.

Step-by-step explanation:

The question asks to find the derivative of y with respect to x, given the equation x(8ⁿ) = 4x⁵ + 3y⁴. To solve this, we need to implicitly differentiate both sides of the equation with respect to x.

Let's differentiate the given function:

1. First, apply the product rule to x times 8ⁿ, which gives us dx/dx · 8ⁿ + x · d/dx(8ⁿ).
2. Then, differentiate the right side of the equation, which is straightforward.
3. Finally, solve for {d y}/{d x} since the question asks for this derivative.

However, without more context or a specific method for differentiation, we can't provide an exact numerical answer.