Differenciate x^6 with respect to 1/√x

1 answer:

7 0

$DIFFERENTIATION \\ \\ \\ Let \: u \: = \: {x}^{6} \: \: \: and \: v = \frac{1}{ \sqrt{x} } \\ \\ Then \: , \: \frac{du}{dx} \: = \: 6 {x}^{5} \\ \\ and \: \: \: \: \: \: \frac{dv}{dx} \: \: = \: \frac{ - 1}{2x \sqrt{x} } \\ \\ \\ Therefore \: , \\ \\ \\ \frac{du}{dv} \: = \: \frac{ \frac{du}{dx} }{ \frac{dv}{dx} } \: = \: \frac{6 {x}^{5} }{ \frac{ - 1}{2x \sqrt{x} } } \\ \\ \frac{du}{dv} \: = \: \frac{ \frac{du}{dx} }{ \frac{dv}{dx} } \: = - 12 {x}^{ \frac{13}{2} } \\ \\ \\ \frac{du}{dv} \: = - 12 {x}^{ \frac{13}{2} } \: \: \: \: \: \: \: \: Ans.$

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The average rate of change between x=1 and x=4 is 22. The average rate of changed describes the rate at which miles is changing with respect the the change in hour.

Part C: Answer=10hours. Since by now we know both the slope and the y-intercept, lets write the equation in slope intercept form and then solve for 225 miles.

y=mx+b
y=22x+5
225=22x+5
subtract 5 from both sides
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