Derivatives of functions given implicitly

For each of the following curves, use implicit differentiation to find $dy/dx$ and determine the equation of the tangent line at the given point.

1. $x^3 - y^3 = 6xy$,  $(-3,3)$

2. $\sin(y) + y = x^3 + x$,  $(0,0)$

3. $3x e^{-xy} = y^2$,  $(0.619061,1)$

Implicit differentiation enables us a different perspective from which to see why the rule $\frac{d}{dx} [a^x] = a^x \ln(a)$ holds, if we assume that $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$. This exercise leads you through the key steps to do so.

1. Let $y = a^x$. Rewrite this equation using the natural logarithm function to write $x$ in terms of $y$ (and the constant $a$).

2. Differentiate both sides of the equation you found in (a) with respect to $x$, keeping in mind that $y$ is implicitl…