Differentiate with respect to x and simplify your answer

y= x2 (2x+1)1/2

You'll need to make use of the following three results. If y = x^a then dy/dx = a*x^(a-1). This is valid for any value of a including 1/2. Product rule. Suppose y has the form of one function of x, say x^3, multiplied by another function of x, say sin(x). Then we can write this as: y = u*v Where we have defined u = x^3 and v = sin(x). Then the product rule states that: dy/dx = (du/dx)v + (dv/dx)u That is, take the derivative of each part of the product separately and multiply the result by the other part, and add these together. For a simple example, take the derviative of x^3. We could just do this in one go using (1), but suppose we write x^3 = (x^2)*x. Then, using the product rule with u = x^2 and v = x: dy/dx = [d(x^2)/dx]x + (dx/dx)x^2 dy/dx = 2x^2 + x^2 dy/dx = 3x^2. 3) Chain rule, "function of a function". Look at the second part of the expression you gave. We see that the entire expression 2x+1 is raised to the power 1/2. We could call this the function "to the power 1/2" of the function "2x+1". Hence the "function of a function". To differentiate this, write 2x+1 = f and y = f^1/2. Then: dy/dx = (dy/df)*(df/dx) For dy/df you'll get (1/2)f^(-1/2) using rule 1. You must replace f by its value in terms of x, ie (1/2)*(2x+1)^(-1/2). Meanwhile df/dx is just 2, so dy/dx becomes (2x+1)^(-1/2). Now you need to put all these parts together, using the result at the end of part 3 and applying the product rule to the whole equation. Hope this is clear, there's a lot to take in. James
05 September 2012
Adarsh tarai
15 October 2020
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