Find $$ \frac{d y}{d x} $$ by implicit differentiation. $$ \begin{array}{l} 4 x^{2}-7 y^{2}=1 \ \frac{d y}{d x}= \end{array} $$ Need Help? Road It

Question

UpStudy AI · Accepted Answer

We are given the equation

4x² − 7y² = 1.

Differentiate both sides with respect to x:

d/dx (4x²) − d/dx (7y²) = d/dx (1).

Calculate each derivative:

• d/dx (4x²) = 8x

• d/dx (7y²) = 7 · d/dx (y²) = 7 · (2y · dy/dx) = 14y · dy/dx

• d/dx (1) = 0

Thus, the differentiated equation becomes:

8x − 14y (dy/dx) = 0.

Now, solve for dy/dx:

Subtract 8x from both sides:
    −14y (dy/dx) = −8x

Divide both sides by −14y:
    dy/dx = (−8x)/(−14y) = (8x)/(14y).

Simplify the fraction by dividing numerator and denominator by 2:

dy/dx = (4x)/(7y).

So, the derivative is:

dy/dx = (4x)/(7y).
$$ \frac{d y}{d x} = \frac{4x}{7y} $$

Find \( \frac{d y}{d x} \) by implicit differentiation. \[ \begin{array}{l} 4 x^{2}-7 y^{2}=1 \\ \frac{d y}{d x}= \end{array} \] Need Help? Road It