In calculus,Explicit is a function which the independent variable. The function f explicitly is to provide a preparation for determining the output of the given function y in terms of the input value x: y = f(x). Derivative of an explicit function is called as explicit differentiation. for example y = x3 + 5. The process of finding the differentiation of the independent variable in an explicit function by differentiating each term separately, by expressing the derivative of the independent variable as a symbol, and by solving the resulting expression for the symbol.
Solve explicit differentiation problems:
Let us see some problems and its helps to solve an explicit differentiation.
Solve explicit differentiation problem 1:
Find the differentiation of given explicit function y = x2 - 15x + 3.
Solution:
Given explicit function is y = x2 - 15x + 3.
Differentiation of explicit function is dy/dx = d/dx ( x2 - 15x + 3)
Separate the each term, so, we get
= d/dx (x2) - d/dx (15x) + d/dx (3)
= d/dx (x2) - 15d/dx(x) + d/dx (3).
= 2 x(2-1) - 15 + 0
= 2x - 15
The differentiation of an explicit function is 2x - 15
Solve explicit differentiation problem 2:
Find the differentiation of an explicit function x2 + cot x = - y
Solution:
Given explicit function is x2 + cot x = - y
Multiply by (-1) on both sides, y = - x2 - cot x
Differentiation of an explicit function,
y = - x2 - cot x
dy/dx = d/dx -x2 - d/dx (cot x) .
= - 2x - (-cosec2 x) .
= - 2x + cosec2x
The Differentiation of an explicit function is - 2x + cosec2x
Solve explicit differentiation problem 3:
Find the differentiation of given explicit function 2x2 + y2 = 1
Solution:
Given explicit function is 2x2 + y2 = 1
Subtract 2x2 on both sides.we get,
2x2 + y2 - 2x2 = 1 - 2x2
y 2 = 1 - 2x2
Take square root on both sides, we get
sqrt (y^2) = +- sqrt (1 - 2x^2)
y = +- sqrt (1 - 2x^2) .
Differentiate the function, Let u = 1 - 2x2 and y = sqrt u
(du)/(dx) = - 4x dy/(du) = 1/(2sqrtu)
So, dy/dx = ((dy)/(du)) ((du)/(dx)) .
= 1/(2sqrtu) . (-4x )
= (- 2x) / sqrtu .
Substitute u = y, So we get
= - (2x)/y .
The differentiation of an explicit function is - (2x)/y .