Derivative rules flashcards for calculus practice

Question 1

What does $f'(x)$ represent?

Accepted Answer

$f'(x)$ is the derivative of $f$ at $x$: the instantaneous rate of change of $f$ with respect to $x$ (slope of the tangent line).

Question 2

State the quotient rule.

Accepted Answer

If $h(x)=\frac{f(x)}{g(x)}$ with $g(x)
eq 0$, then $$h'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}.$$

Question 3

Use implicit differentiation to find $\frac{dy}{dx}$ for $x^2+y^2=1$.

Accepted Answer

Differentiate: $2x+2y\,\frac{dy}{dx}=0$, so $$\frac{dy}{dx}=-\frac{x}{y}$$ (where $y
eq 0$).

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