Solved Problems on Total Derivative
Question 1: Given, [Tex]u = sin\frac{x}{y} , x = e^{t}, y= t^{2}, find \frac{du}{dt}[/Tex]as a function of t. Verify your result by direct substitution.
Solution:
We have, [Tex] \frac{du}{dt} = \frac{\partial u }{\partial x}. \frac{\partial x }{\partial t} + \frac{\partial u }{\partial y}. \frac{\partial y }{\partial t}[/Tex]
= [Tex]cos\frac{x}{y} . \frac{1}{y} . _e{t} + (cos\frac{x}{y}) (\frac{-x}{_y{2}^{ }}) .2t[/Tex]
putting values of x and y in the above equations
= [Tex]cos\frac{_e{t}}{_t{2}} . \frac{_e{t}}{_t{2}} -2cos\frac{_e{t}}{_t{2}} . _e{t}._t{3} [/Tex]
[Tex]\frac{du}{dt} = (t-\frac{2}{_t{3}})._e{t}.cos\frac{_e{t}}{_t{2}} [/Tex]
Question 2: Given, f(x,y) = exsiny , x = t3+1 and y = t4+1. Then df/dt at t = 1.
Solution:
Let f(x,y) =exsiny
[Tex]\frac{df}{dt} =\frac{\partial u }{\partial x}. \frac{\partial x }{\partial t} + \frac{\partial u }{\partial y}. \frac{\partial y }{\partial t}[/Tex]
= exsiny.(3t2) + cosy .ex .(4t3)
As we know , x= t3+1 and y= t4+1
x and y values at t =1, x=2 and y=2
[Tex]\frac{df}{dt} = (e^{2})(sin2)(12) + (cos2)(e^{2})(32) [/Tex]
= (2.718)2(0.0349)(12) +(0.9994)(2.718)2(32)
= 238.97
Question 3: If u = x . log(xy) where x3 + y3 + 3xy = 1, find du/dx.
Solution:
We have x3 + y3 + 3xy = 1……….(1)
[Tex]\frac{du}{dx}=\frac{\partial u}{\partial x}.\frac{dy}{dx} +\frac{\partial u}{\partial y}.\frac{dy}{dx}[/Tex]
= [Tex](logxy +1) + \frac{x}{y}.\frac{dy}{dx} ……….(2) [/Tex]
from eq……….(1)
[Tex]\frac {dy}{dx} = -\frac{\frac{∂f}{∂x}} { \frac{∂f}{∂y}}[/Tex]
[Tex]\frac{dy}{dx} = -\frac {(3x^{2} + 3y)}{(3y^{2} + 3x)} [/Tex]
[Tex]= -\frac{(x^{2} +y)}{(y^{2} +x)}t[/Tex]
after putting value in eq (2)
[Tex]\frac{du}{dx} = (logxy +1) – (\frac{x}{y})\frac {(x^{2}+y)}{(y^{2}+x)} .[/Tex]
Total Derivative
Total Derivative of a function measures how that function changes as all of its input variables change. For function f at a point is an approximation near the point of the function w.r.t. (with respect to) its arguments (variables). The total derivative never approximates the function with a single variable if two or more variables are present in the function.
Sometimes, the Total derivative is the same as the partial derivative or ordinary derivative of the function. In this article, we will discuss about total derivative in detail.
Table of Content
- What is a Total Derivative?
- Formula for Total Derivative
- Total Derivative of Composite Function
- Difference Between Total Derivative and Partial Derivative
- Practice
- FAQs