# 2.15: Homework- Examples of the Definition of the Derivative

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1. Simplify each expression involving fractions or rational expressions.
1. $$(x+1) \cdot \left( \frac{1}{3} + \frac{x}{x+1} \right)$$
$$=\frac{x+1}{3} + x = \frac{4x+1}{3}$$
ans
2. $$\cfrac{\cfrac{1}{3} + 1}{1-\cfrac{1}{3}}$$
$$2$$
ans
3. $$\cfrac{x + 1}{\cfrac{1}{x}}$$
$$x^2 + x$$
ans
4. $$\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}$$
$$\frac{1}{2x+1}$$
ans
5. $$\cfrac{\cfrac{2}{x} - \cfrac{1}{x}}{\cfrac{1 - y}{y}}$$
$$\frac{y}{x - xy}$$
ans
2. In each case, use the definition of the derivative to find $$f'(x)$$ (in other words, take the derivative!)
1. $$f(x) = 3x - 5$$
$$3$$
ans
2. $$f(x) = \frac{1}{2} x + 1$$ ans
3. $$f(x) = 2 x^2$$
$$4x$$
ans
4. $$f(x) = (x^2 + x)$$
$$2x + 1$$
ans
5. $$\frac{d}{dx} (e^x)$$ (hint: from yesterday’s homework, we have $$\lim_{h \to 0} \frac{e^h - 1}{h} = 1$$)
$$e^x$$
ans
6. $$f(x) = x^3$$
$$x^3$$
ans
3. In each case, use the definition of the derivative to find $$f'(x)$$ (in other words, take the derivative!). Each of these is like one of the “hard problems” (click here)
1. $$f(x) = 2x^4$$
$$8x^3$$
ans
2. $$f(x) = \sqrt{2x}$$
$$\frac{1}{\sqrt{2x}}$$
ans
3. $$f(x) = \frac{2}{x}$$
$$-\frac{2}{x^2}$$
ans
4. $$f(x) = \sqrt{x+1}$$
$$\frac{1}{2 \sqrt{x+1}}$$
ans
5. $$f(x) = \frac{1}{x+1}$$
$$-\frac{1}{(x+1)^2}$$
ans
6. $$f(x) = \frac{1}{\sqrt{x}}$$
$$\frac{-1}{2 x \sqrt{x}}$$
ans
4. Recall the derivative of $$f(x) = x^2$$ is given by $$2x$$.
1. Show that the derivative of $$g(x) = x^2+1$$ is $$2x$$ using the definition of the derivative. Can you find an intuitive reason why $$f(x)$$ and $$g(x)$$ would have the same derivative?
Adding a constant moves the curve up or down, but that shift does not affect the slope of the tangent line
ans
2. Find another function whose derivative is $$2x$$, other than $$f(x)$$ and $$g(x)$$.
$$x^2 + c$$ for any value $$c$$
ans

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