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April 2018
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Problem:
The method of power substitution assumes that you are familiar with the method of ordinary usubstitution and the use of differential notation. Integrate the following using the method of substitution.
\[\int \frac{1}{1 + {\sqrt{x}}} dx \]
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The following problem may require the use of the following basic trigonometry derivatives. Also, the derivative, d/dx, may be represented by D.
\[\frac{d}{dx}sinx = cosx\]
\[\frac{d}{dx}cosx = sinx\]
\[sin^{2}x + cos^{2}x = 1\]
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Note the following properties of Natural Logarithms.
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DIFFERENTIATION USING THE QUOTIENT RULE
The following problem requires the use of the quotient rule. The derivative of a function h(x) may be denoted by D. The quotient rule is a formal rule for differentiating problems where one function is divided by another. It follows from the limit definition of derivative and is given by:
\[D\frac{f(x)}{g(x)} = \frac{ g(x) D(f(x))  f(x) Dg(x)}{(g(x))^2}\]
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DIFFERENTIATION USING THE PRODUCT RULE
The following problem requires the use of the product rule. In the following discussion and solution the derivative of a function h(x) will be denoted by or h'(x) . The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by
D{f(x)g(x)} = f(x)g^'(x) + f^'(x)g(x)\]
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DIFFERENTIATION USING THE CHAIN RULE
The following problem requires the use of the chain rule. The chain rule is a rule for differentiating compositions of functions. In the following discussion and solution the derivative of a function h(x) may be denoted by h'(x) . The chain rule states formally that
\[ \frac{d}{dx}f(g(x)) = (\frac{d}{dx}(f(x)))(g(x)) \frac{d}{dx}g(x) \]
