§2.1 Elementary derivative rules (continued)

Constant Multiples and Sums of Functions

Let $g(x) = 14e^x + 3x^5 - x$. Which expression is $g'(x)$?

For $f(x) = x^{5/3} - x^4 + 2^x$, which expression is $f'(x)$?

For $h(z) = \sqrt{z} + \frac{1}{z^4} + 5^z$, which expression is $h'(z)$?

Let $r(t) = \sqrt{53}\, t^7 - \pi e^t + e^4$. Which expression is $\frac{dr}{dt}$?

To differentiate $s(y) = (y^2 + 1)(y^2 - 1)$, the most efficient first step is to

For $p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12$, compute $p'(1)$.

For $h(z) = \sqrt{z} + \frac{1}{z}$, find the slope of the tangent line at $z = 4$, expressed as a fraction.

A cell population in millions is $P(t) = 2(1.37)^t + 32$ with $t$ in days. Compute $P'(4)$, rounded to three decimals, in millions of cells per day.

For the same population $P(t) = 2(1.37)^t + 32$, the growth rate itself is increasing on day $4$.

For $p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12$, the tangent line at $a = -1$ is

Asking for the slope of a tangent line and asking for the equation of the tangent line are two different requests.

The derivative of $e^4$ with respect to $x$ equals zero.