Question

Let's find the derivatives for both parts of the question:

a. If y=5u2+3u−1 and u=x2+5 /18, we want to find dx/dy when x=2.

Answer 1

The question requires us to use calculus to find dx/dy given that y=5u^2+3u-1 and u=(x^2+5)/18. By applying derivative chain rule, we first find dy/du and du/dx, then multiply them to get dy/dx, and finally reciprocate to get dx/dy at x=2.

Step-by-step explanation:

The problem involves calculus and specifically chain rule of differentiation. To find dx/dy, we need to find dy/dx first and then take the reciprocal. Given that y=5u2+3u-1 and u=(x2+5)/18, we will first differentiate y with respect to u, and then differentiate u with respect to x. After calculating dy/du and du/dx, we multiply them to get dy/dx. The reciprocal of this result will give us dx/dy.

To calculate dy/du, apply the power rule to each term:

1. dy/du = d/dx(5u2) + d/du(3u) - d/du(1),
2. dy/du = 10u + 3.

Then, calculate du/dx:

1. du/dx = d/dx((x2+5)/18),
2. du/dx = (1/9)x.

Now, multiply dy/du by du/dx to find dy/dx.

Finally, take the reciprocal of dy/dx to find dx/dy. Since we only need dx/dy at x=2, substitute the value of u when x=2 into the expression for dy/dx, then take the reciprocal.