# 6.5: Derivatives of Functions Given Implicitely

- Page ID
- 106372

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## Supplemental videos

## Homework Exercises 6.5

**1. **Find \(\frac{dy}{dx}\) for the implicit function:

\(x^2y-x-5y-11=0\)

**2. **Find the slope of the tangent line to the implicit function of *x* at the point (1, 5). Use Desmos to sketch a graph of the curve and the tangent line to show that your answer makes sense.

\(x^3+xy+y^2=31\)

**3. **Find the equation of the tangent line to the implicit function of *x* at the point (4, 1).

\(3xy^3+xy=16\)

**4a. **Find \(\frac{dy}{dx}\) for the implicit function:

\(2y^3+y^2-y^5 = x^4-2x^3 + x^2\)

**b. **Find where \(\frac{dy}{dx} = 0\) to find the *x*-coordinates where there are horizontal tangent lines. You should be able to solve for the zeros by factoring the numerator.

**c. **Use Desmos to sketch a graph of the curve to show there are nine points where there are horizontal tangent lines.

**5a. **For the curve given by \(\sin(x+y) + \cos(x-y) = 1\), find \(\frac{dy}{dx}\).

**b. **Use Desmos to graph this implicit function. (Isn't it interesting??) Then zoom in near the point \((\frac{\pi}{2}, \frac{\pi}{2})\). Give a rough estimate of \(\frac{dy}{dx}\bigg|_{(\frac{\pi}{2}, \frac{\pi}{2})}\).

**c**. Find the exact value of the slope using your derivative function and knowledge of the unit circle. Then find the equation of the tangent line and graph this along with the implicit function on desmos.