# Calculator lab - nDeriv

Today we are exploring what graphing calculators can do with derivatives.

TI-84 command: MATH → nDeriv()

Recall our work with $$f(x) = x^2$$. Try f'(1) = nDeriv(x^2, x, 1) or $$\frac{d}{d[]} ( [] ) |_{x=[]}$$.

For each of the situations below, compute $$f'(a)$$ for $$a \in \{ 1, 2, 3, 4, 5 \}$$. What patterns do you notice?

1. Constants: Let $$f(x) = c$$, where $$c$$ is a different constant choosen by each group member.
2. Squares: Let $$f(x) = x^2$$. If possible, write each $$f'(a)$$ showing a common factor. For example, 6 and 10 have a common factor of 2 so could be written as: $$2 \cdot 3$$ and $$2 \cdot 5$$.
3. Cubics: Let $$f(x) = x^3$$. If possible, write each $$f'(a)$$ showing a common factor.
4. Quartics: Let $$f(x) = x^4$$. If possible, write each $$f'(a)$$ showing a common factor.
5. Coefficients:Let $$f(x) = c \cdot x^2$$. The group should agree on a constant $$c$$ (please do not use 0 or 1). (Hint: It may help to compare to "squares" above.)
6. e: Let $$g(x) = e^x$$ . Compute $$g(a)$$ and $$g'(a)$$.
7. Natural log:Let $$h(x) = \ln(x)$$. Compute $$h'(a)$$. (Hint: It may help to use fraction form.)

## Challenge 1

Use the definition of a derivative to verify the pattern for $$f(x) = x^3$$.

## Challenge 2

On the same graph, plot $$x^2$$ and $$\frac{d}{dx} (x^2)$$ using nDeriv(). How did you do this?