**Title: First-Order Partial Derivatives of a Function** --- **Problem Statement:** Compute the first-order partial derivatives of the function: \[ z = \frac{14x}{\sqrt{x^2 + y^2}} \] (Use symbolic notation and fractions where needed.) --- **Partial Derivatives:** \[ \frac{\partial z}{\partial x} = \boxed{\phantom{input}} \] \[ \frac{\partial z}{\partial y} = \boxed{\phantom{input}} \] --- In this problem, we are asked to compute the first-order partial derivatives with respect to \(x\) and \(y\) for the given function \(z\). The function \(z\) is given as a quotient involving the variables \(x\) and \(y\), which is nested inside a square root in the denominator. To solve for the partial derivatives, you would typically: 1. **Differentiate \(z\) with respect to \(x\)**: This involves applying the quotient rule and the chain rule. 2. **Differentiate \(z\) with respect to \(y\)**: Again, the quotient rule and the chain rule come into play. By practicing these differentiations, students will gain a deeper understanding of how to handle partial derivatives for more complex functions. **Note:** Make sure to follow each step of differentiation carefully, checking your work for accuracy.

is (2,4),(3,4),(4,4) a function

Suppose the demand for Colin's kidney pies is linear in the price P. Further, assume that he can sell 100 pies when the price is $5.00 and 65 pies when the price is $10.00. (a) Determine the demand N (number of pies sold) as a function of the price P (in dollars). (Express numbers in exact form. Use symbolic notation and fractions where needed.) N(P) = (b) What is the slope of the graph of N(P)? (Give an exact answer. Use symbolic notation and fractions where needed.) slope: pies/dollar

Write the function as a composition of two functions. 2 3-(x²+6x)³-3(x²-6x)³+6 y= (x²+6x)³-3(x²+6x) ³ +6 Choose the correct answer below. ○ A. If f(x) = x²+6x and g(x)=x ов. о с. 1 2 1 2/3 3x3 +6, then y f[g(x)]. 2 If f(x)=x -3x3 +6 and g(x)=x+6x, then y = f[g(x)]. 3 3 2 2 If f(x)=x³-3x² +6 and g(x) = x²+6x, then y = f[g(x)]. 1 OD. If f(x) = 6x³ 2 -18x3 +6 and g(x) = x², then y = f[g(x)].