Derivatives
We will cover following topics
Derivative of a function of single variable
The derivative of \(f(x)\) is defined by \(f'(x)=\lim _{h \rightarrow 0} \dfrac{f(x+h)-f(x)}{h}\). The following notations are used to denote the derivative of a function \(y=f(x)\): \(y'\), \(f'(x)\), \(\dfrac{df}{dx}\), \(\dfrac{dy}{dx}\) and \(Df(x)\).
Application of Derivatives of a Single Variable
Maxima and Minima
The maxima and minima of a function \(f\) can be obtained by equating its derivative to 0 and then check for second-order derivative at the roots.
(i) For a maxima, \(\dfrac{d y}{d t}=0\) and \(\dfrac{d^{2} y}{d t^{2}}< 0\)
(ii) For a minima, \(\dfrac{d y}{d t}=0\) and \(\dfrac{d^{2} y}{d t^{2}}> 0\)
(iii) For a saddle point, \(\dfrac{d y}{d t}=0\) and \(\dfrac{d^{2} y}{d t^{2}}= 0\)
Rolle’s Theorem and MVT
For a function \(f\) which is continuous on \([a,b]\) and differentiable on \((a,b)\), where \(a< b\), we state the following two theorems:
Rolle’s Theorem: If \(f(a)=f(b)\). Then, according to Rolle’s Theorem, \(\exists \text { } c\) such that \(c \in(a, b)\) and \(f'(c)=0\).
Mean Value Theorem: There exists \(c \in (a,b)\) such that \(f'(c)= \left( \dfrac{f(b)-f(a)}{b-a} \right)\)
Partial Derivatives
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
The partial derivative of a function \(f(x, y, \ldots)\) with respect to the variable \(x\) is variously denoted by: \(f_{x}^{\prime}\), \(f_{\pi}\), \(\partial_{x} f\), \(D_{x} f\), \(D_{1} f\), \(\dfrac{\partial}{\partial x} f\), or \(\dfrac{\partial f}{\partial x}\).
Applications of Partial Derivatives
Maxima and Minima
Lagrange’s Method of Multipliers: Lagrange’s method of multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. We can be provided with a single constraint or multiple constraints.
Jacoabian
The Jacobian martix is the matrix of all first-order partial derivatives of a vector-valued function. If the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.
Approximations
Let \(z=f(x,y)\) be a function that possesses continuous partial derivatives wrt \(x\) and \(y\). Then, the approximate change \(dz\) in \(z\) corresponding to the small changes \(\delta x\) and \(\delta y\) in \(x\), \(y\) is given by:
\[dz = \dfrac{\partial z}{\partial x} d x+\dfrac{\partial z}{\partial y} d y\]PYQs
Derivative of a function of single variable
1) Find the supremum and infimum of \(\dfrac{x}{\sin x}\) on the interval \(\left( 0, \dfrac{\pi}{2} \right]\)
[2017, 10M]
2) For the function \(f:(0, \infty) \rightarrow R\) given by \(f(x)=x^{2} \sin \dfrac{1}{x}\), \(0< x < \infty\), show that there is a differentiable function \(g : R \rightarrow R\) that extends \(f\).
[2016, 10M]
3) State Rolle’s theorem. Use it to prove that between two roots of \(e^{x} \cos x=1\), there will be a root of \(e^{x} \sin x=1\).
[2009, 12M]
4) For \(x>0\), show \(\dfrac{x}{1+x}<log(1+x)<x\).
[2008, 6M]
5) Using Lagrange’s mean value theorem, show that \(\vert \cos b-\cos a \vert \leq \vert b-a \vert\).
[2007, 12M]
6) A twice differentiable function \(f\) is such that \(f(a)=0\) and \(f(c)>0\) for \(a < c < b\). Prove that there is at least one value \(\xi, a < \xi < b\) for which \(f^{\prime \prime}(\xi) < 0\).
[2006, 20M]
7) If \(f^{\prime}\) and \(g^{\prime}\) exist for every \(x \in [a, b]\) and if \(g^{\prime}(x)\) does not vanish anywhere in \((a, b)\), show that there exists \(c\) in \((a, b)\) such that \(\dfrac{f(c)-f(a)}{g(b)-g(c)}=\dfrac{f^{\prime}(c)}{g^{\prime}(c)}\)
[2005, 30M]
Partial Derivatives
1) Show that the function
\[f(x, y)=\left\{\begin{array}{ll}{\dfrac{x^{2}-y^{2}}{x-y},} & {(x, y ) \neq(1,-1),(1,1)} \\ {0} & {, (x, y )=(1,1), (1,-1) }\end{array}\right.\]is continuous and differentiable at \((1,-1)\).
[2019, 10M]
2) Obtain \(\dfrac{\partial^{2} f(0,0)}{\partial x \partial y}\) and \(\dfrac{\partial^{2} f(0,0)}{\partial y \partial x}\) for the function
\[f(x, y)=\left\{\begin{array}{c}{\dfrac{x y\left(3 x^{2}-2 y^{2}\right)}{x^{2}+y^{2}},(x, y) \neq(0,0)} \\ {0 \quad,(x, y) = (0,0)}\end{array}\right.\]Also, discuss the continuity \(\dfrac{\partial^{2} f}{\partial x \partial y}\) and \(\dfrac{\partial^{2} f}{\partial y \partial x}\) of \(f\) at \((0,0)\).
[2014, 15M]
3) Let \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{(x+y)^{2}}{x^{2}+y^{2}},} & {\text { if }(x, y) \neq(0,0)} \\ {1,} & {\text { if }(x, y)=(0,0)}\end{array}\right.\).
Show that \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) exist at \((0,0)\) though \(f(x, y)\) is not continuous at \((0,0)\).
[2012, 15M]
4) Show that the function given by \(f(x, y)=\left \{ \begin{array}{ll}{\dfrac{x y}{x^{2}+2 y^{2}}} & {(x, y) \neq(0,0)} \\ {0} & {(x, y)=(0,0)}\end{array}\right.\) is not continuous at \((0,0)\) but its partial derivatives \(f_{x}\) and \(f_{y}\) exist at \((0,0)\).
[2007, 12M]
5) Show that the function given by \(f(x, y)=\left \{ \begin{array}{c}{\dfrac{x^{3}+2 y^{3}}{x^{2}+y^{2}},(x, y) \neq(0,0)} \\ {0 \quad,(x, y)=(0,0)}\end{array}\right.\)
i) is continuous at \((0,0)\)
ii) possesses partial derivative \(f_{x}(0,0)\) and \(f_{y}(0,0)\)
[2006, 20M]
Applications of Partial Derivatives
1) Using differentials, find an approximate value of \(f(4.1,4.9)\) where
\[f(x,y)=(x^3+x^2y)^{\dfrac{1}{2}}\][2019, 15M]
2) Find the maximum value of \(f(x,y,z)=x^2y^2z^2\) subject to the subsidiary condition \(x^2+y^2+z^2=c^2\), \((x,y,z>0)\).
[2019, 15M]
3) Find the relative maximum minimum values of the function \(f(x, y)=x^{4}+y^{4}-2 x^{2}+4 x y-2 y^{2}\).
[2016, 15M]
4) Find the absolute maximum and minimum values of the function \(f(x, y)=x^{2}+3 y^{2}-y\) over the region \(x^{2}+2 y^{2} \leq 1\).
[2015, 15M]
5) Find the minimum value of \(x^{2}+y^{2}+z^{2}\) subject to the condition \(x y z=a^{3}\) by the method of Lagrange multipliers.
[2014, 15M]
6) Let \(f(x,y)= y^2+4xy+3x^2+x^3+1\). At what points will \(f(x,y)\) have a maximum or minimum?
[2013, 10M]
7) Find the minimum distance of the line given by the planes \(3 x+4 y+5 z=7\) and \(x-z=9\) and from the origin, by the method of Lagrange’s multipliers.
[2012, 15M]
8) Find the shortest distance from the origin \((0,0)\) to the hyperbola \(x^{2}+8 x y+7 y^{2}=225\).
[2011, 15M]
9) If \(u\), \(v\), \(w\) are the roots of the equation in \(\lambda\) and \(\dfrac{x}{a+\lambda}+\dfrac{y}{b+\lambda}+\dfrac{z}{c+\lambda}=1\), evaluate \(\dfrac{\partial(x, y, z)}{\partial(u, v, w)}\).
[2005, 12M]
10) If \((x, y, z)\) be the lengths of perpendiculars drawn from any interior point \(P\) of triangle \(A B C\) on the sides \(B C\), \(C A\) and \(A B\) respectively, then find the minimum value of \(x^{2}+y^{2}+z^{2}\), the sides of the triangle \(A B C\) being \(a\), \(b\), \(c\).
[2004, 20M]
11) Show that the maximum value of \(x^{2} y^{2} z^{2}\) subject to condition \(x^{2}+y^{2}+z^{2}=c^{2}\) is \(\dfrac{c^6}{27}\). Interpret the result.
[2003, 20M]
12) Obtain the maxima and minima of \(x^{2}+y^{2}+z^{2}-y z-z x-x y\) subject to condition \(x^{2}+y^{2}+z^{2}-2 x+2 y+6 z+9=0\).
[2002, 25M]
13) Show that \(U=x y+y z+z x\) has a maximum value when the three variables are connected by the relation \(a x+b y+c z=1\) and \(a\), \(b\), \(c\) are positive constants satisfying the condition \(2(a b+b c+c a)>\left(a^{2}+b^{2}+c^{2}\right)\).
[2001, 20M]