2.15: Homework- Examples of the Definition of the Derivative
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Simplify each expression involving fractions or rational expressions.
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\((x+1) \cdot \left( \frac{1}{3} + \frac{x}{x+1} \right)\)
\(=\frac{x+1}{3} + x = \frac{4x+1}{3}\)ans
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\(\cfrac{\cfrac{1}{3} + 1}{1-\cfrac{1}{3}}\)
\(2\)ans
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\(\cfrac{x + 1}{\cfrac{1}{x}}\)
\(x^2 + x\)ans
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\(\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}\)
\(\frac{1}{2x+1}\)ans
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\(\cfrac{\cfrac{2}{x} - \cfrac{1}{x}}{\cfrac{1 - y}{y}}\)
\(\frac{y}{x - xy}\)ans
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\((x+1) \cdot \left( \frac{1}{3} + \frac{x}{x+1} \right)\)
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In each case, use the definition of the derivative to find \(f'(x)\) (in other words, take the derivative!)
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\(f(x) = 3x - 5 \)
\(3\)ans
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\(f(x) = \frac{1}{2} x + 1 \)
ans
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\(f(x) = 2 x^2\)
\(4x\)ans
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\(f(x) = (x^2 + x)\)
\(2x + 1\)ans
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\(\frac{d}{dx} (e^x)\) (hint: from yesterday’s homework, we have \(\lim_{h \to 0} \frac{e^h - 1}{h} = 1\))
\(e^x\)ans
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\(f(x) = x^3\)
\(x^3\)ans
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\(f(x) = 3x - 5 \)
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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)
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\(f(x) = 2x^4\)
\(8x^3\)ans
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\(f(x) = \sqrt{2x}\)
\(\frac{1}{\sqrt{2x}}\)ans
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\(f(x) = \frac{2}{x} \)
\(-\frac{2}{x^2}\)ans
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\(f(x) = \sqrt{x+1}\)
\(\frac{1}{2 \sqrt{x+1}}\)ans
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\(f(x) = \frac{1}{x+1} \)
\(-\frac{1}{(x+1)^2}\)ans
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\(f(x) = \frac{1}{\sqrt{x}}\)
\(\frac{-1}{2 x \sqrt{x}}\)ans
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\(f(x) = 2x^4\)
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Recall the derivative of \(f(x) = x^2\) is given by \(2x\).
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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 lineans
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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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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?