Differentiation Using Substitution





In this page differentiation using substitution we are going to see some example problems to understand this topic better.

Example 1:

Differentiate sin⁻¹ (3x - 4x³) with respect to x

Solution:

Let y = sin⁻¹ (3x - 4x³). First let us consider (3x - 4x³). If we plug sin θ instead of x in the given function we will get 3 sin θ - 4 sin ³ θ. This the formula for sin 3θ.

So put x = sin θ

          θ = sin⁻¹ x

y = sin⁻¹ [ 3(sin θ) - 4(sin θ)³ ]

y = sin⁻¹ [ 3 Sin θ - 4 sin³ θ ]

y = sin⁻¹ [  Sin 3θ ]

y = 3 θ

y = 3 sin⁻¹ x

differentiating with respect to x on both sides

dy/dx = 3(1/√(1 - x²))

dy/dx = 3/√(1 - x²)


Example 2:

Differentiate tan⁻¹ [ (1+x²)/(1-x²) ] with respect to x

Solution:

Take y = tan⁻¹ [ (1+x²)/(1-x²) ]

Let t = [ (1+x²)/(1-x²) ]

So the function has become y = tan⁻¹ t

To differentiate this function with respect to x we have to write the formula required.

dy/dx = (dy/du) x (dt/dx)

t = [ (1+x²)/(1-x²) ]

For differentiating this we have to apply the quotient rule. So u = (1+x²) and v = (1-x²)

u' = 0 + 2x             v' = 0 - 2x

u' = 2x                  v' = -2x

(U/V)' =  [VU' - UV'] /V²

dt/dx = [(1-x²) 2x - (1+x²)(-2x)]/(1-x²)²

dt/dx = [(2x - 2x³) - (-2x - 2x³)]/(1-x²)²

dt/dx = [(2x - 2x³ + 2x + 2x³)]/(1-x²)²

dt/dx = [4x/(1-x²)²]

y = tan⁻¹ t

dy/dt = 1/(1+t²)

dy/dt = 1/(1+[(1+x²)/(1-x²)]²)

dy/dt = 1/(1+[(1+x²)²/(1-x²)²])

dy/dt = 1/[(1-x²)]²+[(1+x²)²]/(1-x²)]²)

dy/dt = (1-x²)]²/[(1-x²)]²+[(1+x²)²]

dy/dt = (1-x²)]²/[(1 + x⁴ - 2x²)+(1+ x⁴ + 2x²)]

dy/dt = (1-x²)²/(2 + 2x⁴)

dy/dx = (dy/dt) x  (dt/dx)

        = (1-x²)²/(2 + 2x⁴) x  4x/(1-x²)²

         = 4x/(2 + 2x⁴)

         = 4x/2(1+ x⁴)

dy/dx = 2x/(1+ x⁴)    differentiation using substitution


Related Topics

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

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Differentiation Using Substitution to First Principles