Applying the power rule to an exponential

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

1. 01

   Which of the following is the derivative of $h(x) = x^{2}e^{x}$?

   ·Exponential and trigonometric functions·Introductory

2. 02

   Let $f(x) = x^{e} + e^{x} + e^{e}$ for $x > 0$. Find $f'(x)$.

   ·Exponential and trigonometric functions·Introductory

3. 03

   For a base $a > 0$, the slope of $y = a^{x}$ at $x = 0$ is the limit below. $\displaystyle \lim_{h \to 0} \frac{a^{h} - 1}{h}$

   ·The definition of the derivative·Standard

4. 04

   Use $\lim_{h \to 0} \dfrac{e^{h} - 1}{h} = 1$ to evaluate the limit below. $\displaystyle \lim_{h \to 0} \frac{e^{2 + h} - e^{2}}{h}$

   ·The definition of the derivative·Standard

5. 05

   A bacterial culture contains $N(t) = 500\cdot 3^{t}$ cells $t$ hours after it is started. How fast is the culture growing at $t = 2$?

   ·Exponential and trigonometric functions·Standard

6. 06

   From the definition of the derivative, $\dfrac{d}{dx}a^{x} = C(a)\,a^{x}$ for every base $a > 0$, where $C(a) = \displaystyle\lim_{h \to 0} \dfrac{a^{h} - 1}{h}$. A calculator estimates $C(3) \approx 1.0986$. Without estimating another limit, find the slope of $y = 9^{x}$ at $x = \tfrac{1}{2}$.

   ·Exponential and trigonometric functions·Standard