Calculus: Analytic Geometry and Calculus, with Vectors

166 Functions, limits, derivatives Theorem 3.65 (chain rule) If f and g are functions such that g is differentiable at x and f i ...

3.6 The chain rule and differentiation of elementary functions 167 and all others obtainable by making "finite combinations" of ...

168 Functions, limits, derivatives to obtain TXdlog lxi = d log (-x) -TX lx d(dxx) x Thus we can extend the two formulas in (3.6 ...

3.6 The chain rule and differentiation of elementary functions 169 Problems 3.69 1 Calculate f'(xo) and write the equation of th ...

170 Functions, limits, derivatives 4 Each of the formulas y = x2 sin x, y = xex can be read "y equals u times v." Do this and ob ...

3.6 The chain rule and differentiation of elementary functions^171 10 Supposing that a and w (omega, to keep physicists and engi ...

172 Functions, limits, derivatives 15 Supposing that y is a differentiable function of x for which x2 + xy(x) + [y(x)]2 = 3, app ...

3.6 The chain rule and differentiation of elementary functions 173 we can differentiate with respect to x with the aid of the ch ...

174 Show that if (1) holds, then (2) (3) (4) Functions, limits, derivatives Hn(x)e ax2l2 = (-1)n E ax2f2 dx [-axH,(x) + H'(x)le ...

3.6 The chain rule and differentiation of elementary functions 175 26 The preceding problem involved three functions and the New ...

176 Functions, limits, derivatives Show that, when x 0 0, y'(x) = -qx"-q-1 cos I + pxP-1 sin x5 Tell why this formula cannot be ...

3.7 Rates, velocities 177 disastrous loss of meaning, be abbreviated to the forms (3.72) Average rate = difference quotient = Dy ...

178 Functions, limits, derivatives buzzes through space E3. While other tactics are both possible and use- ful, we suppose that ...

3.7 Rates, velocities 179 The scalar components of the velocity v or v(t) are sometimes denoted by the symbols v., v, v, so that ...

180 Functions, limits, derivatives workaday world, we need still another definition. When we say thata moving body is, at time t ...

3.7 Rates, velocities 181 Supposing that 0 < t < t + At, work out a formula for the average speed of the stone over the ti ...

182 Functions, limits, derivatives current I in the circuit containing the capacitor. Find a formula which gives I in terms of t ...

3.7 Rates, velocities 11 This problem involves the uniform circular helical motion of a particle Q in E3 which runs up the helix ...

184 Functions, limits, derivatives 13 While the matter must remain mysterious until some mathematical secrets have been revealed ...

3.7 Rates, velocities 185 17 A particle P moves in E3 in such a way that the vector r running from the origin to P is r = r[sin ...