Differentiation is a method to find rates of change. It is an important topic for the JEE exam. Derivative of a function y = f(x) of a variable x is the rate of change of y with respect to rate of change of x. This article helps you to learn the derivative of a function, standard derivatives, theorems of derivatives, differentiation of implicit function and higher order derivatives along with solved examples.
The differentiation of a function is a way to show the rate of change of a function at a given point. For real-valued functions, it is the slope of the tangent line at a point on a graph.
The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between x0andx1 becomes infinitely small (infinitesimal). The derivative is often written as dxdy.
In mathematical terms,
If f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by
In implicit differentiation, also known as Chain Rule, differentiate both sides of an equation with two given variables by considering one of the variable as a function of the second variable. In short, differentiate the given function with respect to x and solve for dy/dx. Let us have a look on some the examples.
Example 1: If x2 + 2xy + y3 = 4, find dxdy
Solution: Differentiating both sides w.r.t. x, we get dxd(x2)+2dxd(xy)+dxd(y3)=dxd(4)=2x+2xdxdy+2y+3y2.dxdy=0⇒dxdy=(2x+3y2)−2(x+y)
Example 2: Differentiable log sin x w.r.t cosx
Solution: Let u = log sin x and v = cosxdxdu=cotx&dxdv=2cosx−sinx,
Differentiation process can be continued upto nth derivation of a function. Usually we deal with first order and second order derivatives of the functions.
dxdy is the first derivative of y w.r.t x
dx2d2y is the second derivative of y w.r.t x
Similarly, finding the third, fourth, fifth and successive derivatives of any function, say g(x), which are known as higher-order derivatives of g(x).
nth order derivative numerical notation is gn(x) or dxndny
Example: If y=etan−1x,thenprovethat(1+x2)dx2d2y=(1−2x)dxdy
dy/dx = etan−1x1+x21
= 1+x2etan−1x …(i)
d2y/dx2 = (1+x2)2(1+x2)etan−1x1+x21−2xetan−1x
(d2y/dx2 )(1 + x2) = (1+x2)(1−2x)etan−1x
= (1-2x)dy/dx (from eqn (i))
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