# MA 227 Standard C06

Chain Rule

At the end of the course, each student should be able to…

• C06: ChainRule. Apply the multivariable Chain Rule to compute derivatives and find normal vectors.

## C06. Chain Rule

• The single-variable Chain Rule $$\frac{d}{dt}[f(u(x))]=f’(u(x))u’(x)$$ may be generalized for multiple variable functions.
• If $$f(P)$$ is a function of multiple variables and $$\vect{r}(t)$$ is a vector function of equal dimension, then $$\frac{d}{dt}[f(\vect r(t))]=\nabla f(\vect r(t))\cdot\vect r’(t)$$.
• An immediate result is that $$\nabla f(P_0)$$ is normal to the level curve of $$f$$ passing through $$P_0$$.
• Thus $$\<f_x(P_0),f_y(P_0),-1\>$$ is normal to the surface $$z=f(x,y)$$ at $$P_0$$, and $$z=f(P_0)+f_x(P_0)(x-x_0)+f_y(P_0)(y-y_0)$$ is the tangent plane to the surface $$z=f(x,y)$$ at $$P_0$$.
• The total derivative $$\frac{df}{dx}$$ describes the rate of change of $$f$$ with respect to $$x$$ when the other variables of $$f$$ are dependent on $$x$$ as well.
• The chain rule shows us $$\frac{df}{dx}=\nabla f\cdot\frac{d\vect r}{dx}$$.
• For three variables: $$\frac{df}{dx}=\frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}\frac{dy}{dx}+ \frac{\partial f}{\partial z}\frac{dz}{dx}$$.
• Thus if $$f(x,y)=c$$ defines $$y$$ as a differentiable function of $$x$$, then $$\frac{dy}{dx}=-\frac{f_x}{f_y}$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 13.4 (exercises 1-6, 25-32)
• 13.6 (exercises 1-12)