Differentiation Rules: From Basic Derivatives to Variable-Limit Integrals

All flashcards

What is the constant rule?

If $f(x)=c$ for a constant $c$, then $f'(x)=0$ because the function does not change with $x$.

State the linearity rule.

For constants $a,b$ and differentiable functions $f,g$, $(af+bg)'=af'+bg'$. It includes the constant-factor, sum, and difference rules.

How does the product rule work?

For two differentiable factors, $(fg)'=f'g+fg'$. Differentiate one factor at a time, keeping the other factor unchanged in each term.

What is the chain rule?

For a composition $h(x)=f(g(x))$, $h'(x)=f'(g(x))g'(x)$: differentiate the outer function at the inner input, then multiply by the inner derivative.

How is the derivative of an inverse function related to the original derivative?

If $g=f^{-1}$, then $g'(y)=1/f'(g(y))$ wherever the functions are differentiable and $f'(g(y))\ne0$.

Quiz questions

1. For differentiable $f$ and $g$, with $g(x)\ne0$, which expression gives the derivative of $f(x)/g(x)$?Answer: $\frac{f'g-fg'}{g^2}$
2. For $x>0$, what is the derivative of $x^x$?Answer: $x^x(1+\ln x)$
3. If $F(x)=\int_{a(x)}^{b(x)} f(x,t)\,dt$, which formula gives $F'(x)$ under the Leibniz rule?Answer: $f(x,b(x))b'(x)-f(x,a(x))a'(x)+\int_{a(x)}^{b(x)}\frac{\partial f}{\partial x}(x,t)\,dt$
4. What is the general Leibniz rule for the $n$th derivative of a product $f(x)g(x)$?Answer: $\sum_{k=0}^{n}\binom{n}{k}f^{(n-k)}(x)g^{(k)}(x)$
5. For a positive constant $c$, what is the derivative of $c^{ax}$, where $a$ is constant?Answer: $a c^{ax}\ln c$