Let f: R -R and g : R → Rbe some functions, and let x be a vector in R . Suppose that all the com...
Suppose that g is differentiable at x for all x ∈ R. Let f(x) = |g(x)|. Use the Chain Rule to find f′(x).
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
Exercise 1. Do the following: (a) Write a statement defining the Chain Rule for the functions g: R" → Rm and f: RM + RP. Then describe how it works in a paragraph, assuming the reader is a classmate who has been following the course but missed the lecture on Properties of the Derivative. (b) Explain in detail how the Chain Rule you learned in Calculus I, (fog)(x) = f (g(2)).g'(x), is really just the special case of your statement...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
Let V be the vector space consisting of all functions f: R + R satisfying f(x) = a exp(x) +b cos(x) + csin(x) for some real numbers a, b, and c. (The function exp refers to the exponential, exp(22) = e.) Let F be the basis (exp cos sin of V. Let T :V + V be the linear transformation T(f) = f + f' + 2f" (where f' is the derivative of f). You may use the linearity of...
Please prove by setting up the theorem below (Chain Rule) v:RR is continuously differentiable. Define the Suppose that the function function g : R2R by 8(s, t)(s2t, s) for (s, t in R2. Find ag/as(s, t) and ag/at(s, t) Theorem 15.34 The Chain Rule Let O be an open subset of R and suppose that the mapping F:OR is continuously differentiable. Suppose also thatU is an open subset of Rm and that the functiong:u-R is continuously differentiable. Finally, suppose that...
3) Let X, Y be vector fields. For all functions f, define the commutator X, Y]0=X(Y()-Y(X(f). Show that X, Y=Z is a vector field, by verifying that it satisfies sum rule and product rule: Z(f+g)-Z(+Z(g) Zfg)-fZ(g)+gZ(). Extra credit: write /X, YJ in local coordinates.
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...