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To sum a row of NUMBERS, SELECT the cell immediately to the right. AUTOSUM is in two locations: HOME > AutoSum, and Formulas > AutoSum.nation:

Section 14.4 (3/23/08) Chain Rules with two variables Overview: In this section we discuss procedures for differentiating composite functions with two variables. Then we consider second-order and higher-order derivatives of such functions. Topics: • Using the Chain Rule for one variable • The general Chain Rule with two variables • Higher order partial derivatives Using the Chain Rule for one variable Partial derivatives of composite functions of the forms z = F (g(x, y)) can be found directly with the Chain Rule for one variable, as is illustrated in the FOLLOWING three examples. Example 1 Find the x-and y-derivatives of z = (x 2 y 3 + sin x) 10 . Solution To find the x-derivative, we consider y to be constant and apply the one-variable Chain Rule formula d DX(f 10) = 10 f 9 df dx from Section 2.8. We obtain ∂ ∂x[(x 2 y 3 + sin x) 10] = 10(x 2 y 3 + sin x) 9 ∂ ∂x(x 2 y 3 + sin x) = 10(x 2 y 3 + sin x) 9 (2xy 3 + cos x). Similarly, we find the y-derivative by treating x as a constant and using the same one-variable Chain Rule formula with y as variable: ∂ ∂y [(x 2 y 3 + sin x) 10] = 10(x 2 y 3 + sin x) 9 ∂ ∂y (x 2 y 3 + sin x) = 10(x 2 y 3 + sin x) 9 (3x 2 y 2 ). Example 2 The radius (meters) of a spherical balloon is given as a function r = r(P, T) of the atmospheric pressure P (atmospheres) and the temperature T (degrees Celsius). At one moment the radius is ten meters, the rate of change of the radius with respect to atmospheric pressure is −0.01 meters per ATMOSPHERE, and the rate of change of the radius with respect to the temperature is 0.002 meter per degree. What are the rates of change of the volume V = 4 3 πr3 of the balloon with respect to P and T at that time? Solution We first take the P-derivative with T constant and then take the T-derivative with P constant, using the Chain Rule for one variable in each case to differentiate r 3 . We obtain ∂V ∂P = ∂ ∂P 4 3 πr 3 = 1 3 πr 2 ∂r ∂P ∂V ∂P = ∂ ∂T 4 3 πr 3 = 1 3 πr 2 ∂r ∂T . Setting r = 10, ∂r/∂P = −0.01, and ∂r/∂T = 0.002 then gives ∂V ∂P = 1 3 π(102 )(−0.01) = − 1 3 π .= −1.05 cubic meters atmosphere ∂V ∂T = 1 3 π(102 )(0.002) = 1 15π .= 0.21 cubic meters degree . 3 p. 318 (3/23/08) Section 14.4, Chain Rules with two variables Example 3 What are the x- and y-derivatives of z = F(g(x, y)) at x = 5, y = 6 if g(5, 6) = 10, F0 (10) = −7, gx(5, 6) = 3, and gy(5, 6) = 11? Solution By the Chain Rule formula d dt[F (u(t))] = F 0 (u(t)) u 0 (t) for one variable with first x and then y in place of t, we obtain h ∂ ∂x{F(g(x, y))} i x=5,y=6 = F 0 (g(5, 6)) gx(5, 6) = F 0 (10) gx(5, 6) = (−7)(3) = −21 h ∂ ∂y {F(g(x, y))} i x=5,y=6 = F 0 (g(5, 6)) gy(5, 6) = F 0 (10) gy(5, 6) = (−7)(11) = −77. Partial derivatives of composite functions of the forms F(t) = f (x(t), y(t)) and F(s, t) = f (x(s, t), y(s, t)) can be found directly with the Chain Rule for one variable if the “outside” function z = f(x, y) is given in terms of power functions, exponential functions, logarithms, trigonometric functions, and inverse trigonometric functions rather than just by a letter name. This is illustrated in the following example. Example 4 Find the t-derivative of z = f (x(t), y(t)), where f(x, y) = x 5 y 6 , x(t) = e t , and y(t) = √ t. Solution Because f(x, y) is a product of powers of x and y, the composite function f (x(t), y(t)) can be rewritten as a function of t. We obtain f (x(t), y(t)) = [x(t)]5 [y(t)]6 = (e t ) 5 (t 1/2 ) 6 = e 5t t 3 . Then the Product and Chain Rules for one variable GIVE d dt[f (x(t), y(t))] = d dt (e 5t t 3 ) = e 5t d dt (t 3 ) + t 3 d dt (e 5t ) = 3t 2 e 5t + t 3 e 5t d dt (5t) = 3t 2 e 5t + 5t 3 e 5t . The general Chain Rule with two variables We the following general Chain Rule is needed to find derivatives of composite functions in the form z = f(x(t), y(t)) or z = f (x(s, t), y(s, t)) in cases where the outer function f has only a letter name. We begin with functions of the first type. Theorem 1 (The Chain Rule) The t-derivative of the composite function z = f (x(t), y(t)) is d dt[f (x(t), y(t))] = fx (x(t), y(t)) x 0 (t) + fy (x(t), y(t)) y 0 (t). (1) We assume in this theorem and its applications that x = x(t) and y = y(t) have first derivatives at t and that z = f(x, y) has continuous first-order derivatives in an open circle centered at (x(t), y(t)). Learn equation (1) as the following statement: the t-derivative of the composite function equals the x-derivative of the outer function z = f(x, y) at the point (x(t), y(t)) multiplied by the t-derivative of the inner function x = x(t), plus the y-derivative of the outer function at (x(t), y(t)) multipled by the t-derivative of the inner function y = y(Explanation:

In a worksheet, select the cells that you want to format.On the Home tab, in the Alignment group, CLICK Wrap TEXT . Notes: Data in the cell wraps to FIT the column WIDTH, so if you change the column width, data wrapping adjusts AUTOMATICALLY.

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In the Tab Stop Position box ENTER a horizontal ... Repeat steps 5 through 7 to set other BAR positions. ... vertical LINES between the columns, you can FOLLOW these steps:.

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Examples of PERSONAL information are: a person's NAME, ADDRESS, phone NUMBER or email address. a photograph of a person.

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Keyboard COMMAND: CONTROL (Ctrl) + CKeyboard Command: Control (Ctrl) + CTHE COPY command is used for just that - it copies the text or image you have selected and stores is on your virtual clipboard, until it is overwritten by the NEXT "cut" or "copy" command.

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Tabs are a paragraph-formatting feature used to align TEXT. When you PRESS the Tab key, Word inserts a tab character and moves the INSERTION point to the tab setting, called the tab STOP. ... Tabs are set to distribute text evenly between the LEFT and right margins. Word's default tabs are set every half-inch.

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CorelDraw is designed to edit two-dimensional IMAGES such as LOGOS and POSTERS. Corel corporation developed and RELEASED a software program called ✨ CorelDRAW, a vector graphics editor. The software is a robust graphics suite, proving many features for users to edit graphics.Explanation: