IGNOU MMT 4 2026 - Suppose that is given by and is given

Question

Suppose that $f : \mathbb{R} \to \mathbb{R}^2$ is given by $f(t) = (t, t^2)$ and $g : \mathbb{R}^2 \to \mathbb{R}^3$ is given by

$g(x, y) = (x^2, xy, y^2 - x^2)$ . Compute the derivative of $g \circ f$ .

09 Jan 2026

Answer :

Word Count : 299

We are asked to compute the derivative of (g \circ f) manually. Let’s go step by step. We are given: [ f: \mathbb{R} \to \mathbb{R}^2, \quad f(t) = (t, t^2), ] [ g: \mathbb{R}^2 \to \mathbb{R}^3, \quad g(x,y) = (x^2, xy, y^2 - x^2). ] First, compute (g \circ f(t)): [ g(f(t)) = g(t, t^2) = \big( (t)^2, (t)(t^2), (t^2)^2 - (t)^2 \big) ] [ g(f(t)) = (t^2, t^3, t^4 - t^2). ] Now, we differentiate component-wise with respect to (t): [ \frac{d}{dt} g(f(t)) = \frac{d}{dt}(t^2, t^3, t^4 - t^2) ] [ = (2t, 3t^2, 4t^3

____ _____ ______ ___ _______ ___ _____ ____.
_____ ________ _________ ____ ______ ______ ______.
___ _______ _________ ______ _______ __________ ____.
_______ ____ ____ ____ _____ ________ _______ __________ _______ ___.
_____ ___ ___ _________ ________ ___ _______ ______ _________ ____.
____ _________ ______ _______ ___ ______ ___ _________ _______ __________ ___ ________.
__________ ____ __________ _______ ______ __________ ___ _____.
___ _______ ________ ___ _________ ______ ___ __________ _______ ___.
_________ ___ _______ __________ ________ ___.
_____ _____ _______ ________ _________ ___ __________ ________ _____.
______ __________ ______ _____ ______ __________ __________ ________.
_____ __________ ____ __________ _________ _______ _____.
__________ _______ _____ ________ _____ _______ _________ ______.
____ _______ __________ _________ __________ ____ ________ _______ _________ ____.
__________ __________ ________ ___ ________.
_______ ________ ________ _______ __________ __________ _______.
_______ ________ _____ ____ _______ __________ __________ _____ __________.
________ _______ _____ ___ ____ ________ _____ ________ _____.
___ ______ ______ ______ _____ _________.
_______ _____ _______ _________ ___ ___.
_______ _____ ______ _________ ________ __________.
_______ __________ ________ __________ ________ ______ ______.
_______ ______ __________ ___ _______ _____.
______ _____ _____ __________ _____ ______ _____ ________ __________ _________ __________ ______.
__________ _______ ____ ______ ________.
_____ ___ ___ _________ __________.

IGNOU ACADEMY

Your IGNOU journey, made easier.