What is the slope of the tangent line to the curve $y^3-x{y}^2+x^3=5$ at the point $(1,2)$?

For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.

lim (x → −1) (x²/⁹ − 1) / (x + 1)

Given the function $f\left(x\right)=x^2+3x$, what is the slope of the line tangent to the curve at the point $x=2$?

For what value of $x$ is the tangent line to the function $f\left(x\right)=x^3-3x^2+2$ horizontal?

Which of the following is an equation of the line tangent to the graph of $y=\cos x$ at $x=\frac{\pi }{2}$?

What is the slope of the tangent line to the curve $x^2+y^2=2$ at the point $(1,1)$?

Find an equation of the tangent line to the curve $y=e^{x}cos\left(x\right)+sin\left(x\right)$ at the point $(0,1)$.

Given the parametric equations $x=t^2-4$ and $y=t^2-2t$, find the equation of the tangent line to the curve at the point where $t=3$.

Find an equation of the tangent plane to the surface $z=x^2+y^2$ at the point $(1,2,5)$.