§2.7 Derivatives of functions given implicitly

Introduction

Implicit Differentiation

Summary

For the curve given implicitly by $x^3 + y^2 - 2xy = 2$, find the slope of the tangent line at $(-1,1)$.

The curve $x = y^5 - 5y^3 + 4y$ has $\frac{dy}{dx} = \frac{1}{5y^4 - 15y^2 + 4}$. Which statement about this formula is correct?

For the curve $x = y^5 - 5y^3 + 4y$, the tangent line at the point $(0, 1)$ is

For $\sin(y) + y = x^3 + x$ at the point $(0,0)$, the tangent line is

For $x^3 - y^3 = 6xy$, compute $\frac{dy}{dx}$ at the point $(-3, 3)$.

For $x = y^5 - 5y^3 + 4y$, compute $\frac{dy}{dx}$ at $(0, 1)$ as a fraction.

Implicit differentiation proceeds by treating $y$ as a function of $x$ and differentiating both sides of the equation with respect to $x$.