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From 3 equations in 5 variables, we can (theoretically at least) determine 3 variables in terms of the remaining 2. Thus, 3 variables are dependent and 2 are independent. If we were asked to determine @v=@y, we would know that v is a dependent variable and y is an independent variable, but would not know the @v remaining independent variable. However, the particular notation serves to indicate that we are to @y x obtain @v=@y keeping x constant, i.e., x is the other independent variable. (a) Di erentiating the given equations with respect to y, keeping x constant, gives 1 Fu uy Fv vy Fw wy Fy 0 3 2 Gu uy Gv vy Gw wy Gy 0

The numerical value of a derivative at a specific point can be computed several different ways, depending upon how the derivative is computed. The next example illustrates the most common techniques.

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f[x_]= (x2 x + 1)5; f''[1] 30 D[f[x] {x, 2}] /. x 1 , 30 {x, 2}f[x] /. x 1 30 g Derivative[2][f] g[1] 30 f[x_] = x3 g[1] 6

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Here we have de ned a new function, g, as the second derivative of f. If f is changed, g will be the second derivative of the new function. Note the use of here. This is crucial if g is to re ect the change in f. In each of the rst three parts of this example, the second derivative is computed and then x is replaced by 1.

Hu uy Hv vy Hw wy Hy 0 Fw @ F; G; H Gw Hw @ u; y; w @ F; G; H Fw Gw @ u; v; w Hw

Mathematica computes derivatives of combinations of functions, sums, differences, products, quotients, and composites by memorizing the various rules. If we do not define the functions, we can see what the rules are.

Clear[f, g] D[f[x] + g[x], x] f '[x] + g'[x] D[f[x] g[x], x] g[x] f'[x] + f[x] g'[x] D[f[x]/g[x], x]//Together g[x]f'[x] f[x]g'[x] 2 g[x] D[f[g[x]], x] f '[g[x]] g'[x]

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Quotient rule. The derivative of a sum is the sum of the derivatives of its terms. This is the familiar product rule.

Equations (1), (2), and (3) can also be obtained by using di erentials as in Problem 6.31. The Jacobian method is very suggestive for writing results immediately, as seen in this problem and @v Problem 6.31. Thus, observe that in calculating the result is the negative of the quotient of two @y x Jacobians, the numerator containing the independent variable y, the denominator containing the dependent variable v in the same relative positions. Using this scheme, we have @ F; G; H @x @ v; y; u @ F; G; H @v w @ x; y; u @ F; G; H @w @ u; x; v @ F; G; H @u y @ w; x; v

We can use Mathematica to investigate some basic theory from a graphical perspective. Rolle s Theorem guarantees, under certain conditions, the existence of a point where the derivative of a function is 0: Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and suppose f(a) = f(b) = 0. Then there exists a number, c, between a and b, such that f ' (c) = 0. In other words, if a smooth (differentiable) function vanishes (has a value of 0) at two distinct locations, its derivative must vanish somewhere in between.

EXAMPLE 13 Show that the function f ( x ) = ( x 3 + 2 x 2 + 15 x + 2)sin x satisfies Rolle s Theorem on the interval

[0, 1] and find the value of c referred to in the theorem. Since f is the product of a polynomial and a trigonometric sine function, f is continuous and differentiable everywhere. f[x_]=(x3 + 2 x2 + 15 x + 2) Sin[o x]; f[0] 0 f[1] 0 FindRoot[f'[c] 0, {c, 0.5}]

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