Fundamentals of Differentiation

$h''(5)=120t$

$h(5)=604$

$h'(5)=620$

$\Delta t=0$

$\frac{1}{e}$

$\frac{1}{e^2}$

$e$

$e^2$

$x^2+2$

$3x^2+2$

$3x^2$

$6x + 2$

The integral of $f$ from $0$ to $3$

The second derivative of $f$ at $x = 3$

The value of the function $f$ when $x = 3$

The derivative of $f$ at $x = 3$

If $h(x)$ has constants added to $x$ within polynomial expressions covering all $x$-values.

If $h(x)$ involves trigonometric functions like sine and cosine throughout its domain.

If $h(x)$ contains exponential growth components like $e^x$ across its entire range.

If $h(x)$ has discontinuities such as jumps or gaps in its domain.

The average rate of change of the function.

The slope of the function at the point.

The integral of the function over a given interval.

The function's derivative at that point.

Speed increasing, velocity decreasing.

Both speed and velocity are decreasing.

Speed decreasing, velocity increasing.

Both speed and velocity are increasing.

The second derivative test is inconclusive for concavity.

The graph of g is concave up at $x=a$.

The graph of g is concave down at $x=a$.

The graph of g has an inflection point at $x=a$.

A jump discontinuity since L'Hôpital's Rule requires conditions similar to differentiability on both functions near $k$.

An infinite discontinuity as long as both functions approach infinity separately when nearing $k$ from either side.

Any type of discontinuity because all types invalidate L’Hôpital’s Rule which relies strictly on continuity and differentiability around $k$.

A removable discontinuity if $f(k)$ and $g(k)$ are both redefined properly before applying L'Hôpital’s Rule.

$-3\sin(\frac{\pi}{2})$

$27\sin^2(\frac{\pi}{2})\cos(\frac{\pi}{2})$

$-3\cos(\frac{\pi}{2})$

$0$