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# Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function $f(r) = 2 \sqrt {Dr},$ where $r$ is the reproductive rate of individuals and $D$ is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate $r$ and explain its meaning.

## $f^{\prime}(r)=\sqrt{\frac{D}{r}}$

Derivatives

Differentiation

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all right, so this function represents the speed of the wave of the invasive species. Colonising as a function of our the reproductive rate and D is just some constant, and we want to find the derivative. The derivative is going to represent the acceleration of the wave as a function of the reproductive weight rate, because the derivative of speed would be acceleration. So before we find the derivative, let's change the equation to to Times Square root D That should be a two to times square root d. That's the constant part of it. Times are to the 1/2 that's the variable part of it. Now let's find the derivative. So F crime of our is the constant to square root D multiplied by the derivative of art. Of the 1/2 which would be 1/2 are to the negative 1/2 so we can simplify that by multiplying the two in the 1/2. And then we can think of are to the negative 1/2 power as one over square root are and then we might as well combine those into the same square roots. So we have the rate of change of the speed is the square root of D over our

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Derivatives

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