If y = (x – 1)(2x + 3)(sec x); evaluate y'.
Strategy # 1 - multiply out 1st, then differentiate:
y = (2x^2 + x – 3)(sec x) = 2x^2(sec x) + x(sec x) – 3(sec x)
y' = 2x^2(sec x)(tan x) + 4x(sec x) + x(sec x)(tan x) + (sec x) – 3(sec x)(tan x).
Strategy # 2 - use the product rule, then segregate and recombine like terms:
y' = (x – 1)(2x + 3)(sec...
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If the function “cos sqrt(x) dx” has to be integrated, would you consider using the substitution method?
Let w = sqrt(x), dw = dx/2sqrt(x), or 2dw = dx/sqrt(x) . . .
Are we stuck? Typically, we employ the substitution method for integration when we can identify a function and its derivative within the integrand. For example, in integral[cos x sin x dx], we recognize the function...
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The tool of mathematics is easily considered the most exact of the sciences. However, if these usually very precise answers to homework and exam problems were cities or towns on a map, then part of math's beauty is that several roads can lead into it. We have perhaps seen or worked word problems which could be solved using more than one branch of math, for example, calculating the required dimensions...
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