# Differentiation of Functions in Parametric Form

Sometimes

*x*and*y*are given as functions of a single variable, e.g.,*x*=*Ï†*(*t*),*y*=*Ïˆ*(*t*) are two functions and*t*is a variable. In such a case*x*and*y*are called parametric functions or parametric equations and*t*is called the parameter. To find*dy*/*dx*in case of parametric functions, we first obtain the relationship between*x*and*y*by eliminating the parameter*t*and then we differentiable it with respect to*x*. But every time it is not convenient to eliminate the parameter. Therefore,*dy*/*dx*can also be obtained by the following formula: =

# Differentiation using logarithm

If

*y*= or*y*=*f*_{1}(*x*) â‹…*f*_{2}(*x*) â‹…*f*_{3}(*x*) ... or*y*=

then it is convenient to take the logarithm of the function first and then differentiate.

** Note: **Write

*y*= and differentiate easily or if

*y*= [

*f*(

*x*)]

^{ g(x)}, then

*dy*/

*dx*= differential of

*y*treating

*f*(

*x*) as constant + differential of

*y*treating

*g*(

*x*) as constant.

For example, if

*y*= , then we can find by the following steps: = (differential of

*y*keeping base sin*x*as constant) + (differential of*y*keeping power log case*x*as constant)=

# Differentiation of one function w.r.t. other function

Let

*u*=*f*(*x*) and*v*=*g*(*x*) be two functions of*x*. Then to find the derivative of*f*(*x*) w.r.t.*g*(*x*), i.e., to find*du*/*dv*we use the following formula .Thus, to find the derivative of

*f*(*x*) w.r.t.*g*(*x*), we first differentiate both w.r.t.*x*. and then divide the derivative of*f*(*x*) wrt*x*by the derivative of*g*(*x*) w.r.t.*x*# Differentiation of determinants

To differentiate a determinant, we differentiate one row (or column) at a time, keeping others unchanged.

For example, if

Î”(

*x*) =then {Î”(

*x*)} =Also, {Î”(

*x*)} =Similar results hold for the differentiation of determinants of higher order.

# Higher-order derivatives

If

*y*=*y*(*x*), then*dy*/*dx*, the derivative of*y*with respect to*x*, is itself, in general, a function of*x*and can be differentiated again. We call*dy*/*dx*as the first-order derivative of*y*with respect to*x*and the derivatives of*dy*/*dx*w.r.t.*x*as the second-order derivative of*y*w.r.t.*x*and will be denoted by*d*^{2}*y*/*dx*^{2}. Similarly, the derivative of*d*^{2}*y*/*dx*^{2}w.r.t.*x*will be termed as the third-order derivative of*y*w.r.t.*x*and will be denoted by*d*^{3}*y*/*dx*^{3}and so on. The*n*th-order derivative of*y*w.r.t.*x*will be denoted by*d*/^{n}y*dx*.^{n}If

*y*=*f*(*x*), then the other alternative notations for are

*y*

_{1},

*y*

_{2},

*y*

_{3}, â€¦,

*y*

_{n}*y*â€²,

*y*â€²â€²,

*y*â€²â€²â€², â€¦,

*y*

^{(n)}

*f*â€²(

*x*),

*f*â€²â€²(

*x*),

*f*â€²â€²â€²(

*x*), â€¦,

*f*(

^{n}*x*)

The values of these derivatives at

*x*=*a*are denoted by*y*(

_{n}*a*),

*y*(

^{n}*a*),

*D*(

^{n}y*a*),

*f*(

^{n}*a*) or