Derivative Concepts, Rules, Multivariable Change, and Antiderivatives

All flashcards

Derivative at a point

The limit of the difference quotient as the input change approaches zero, if that limit exists.

Differentiability and continuity

Differentiability at a point implies continuity there, but continuity alone does not imply differentiability.

Chain rule

For a composition, differentiate the outer function at the inner function and multiply by the inner derivative.

Gradient

The vector of first partial derivatives of a real-valued function of several variables.

Antiderivative

A function $A$ whose derivative equals a given function $f$, so that $A'=f$.

Quiz questions

1. For differentiable functions $f$ and $g$, which expression gives the derivative of their product?Answer: $(fg)'=f'g+fg'$
2. What is the derivative of $f(x)=\sin(x^2)$?Answer: $2x\cos(x^2)$
3. Why is $f(x)=|x|$ not differentiable at $x=0$?Answer: Its left and right slopes at zero are different.
4. If the partial derivatives of $f$ are continuous and $\mathbf{u}$ is a unit direction vector, how is the directional derivative expressed?Answer: $D_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}$
5. For a map $F:\mathbb{R}^n\to\mathbb{R}^m$, what does the entry in row $i$, column $j$ of its Jacobian represent?Answer: $\frac{\partial F_i}{\partial x_j}$