Test: Definite Integration: Leibnitz Theorem (3 Oct) - JEE MCQ

The key here is a simple manipulation and application of the Leibniz rule.
Rewriting the given function as
y(1 + x²) = sin(x)…….(1)
The Leibniz rule for two functions is given by

Differentiating expression (1) in accordance to Leibniz rule (upto the hundredth derivative) we have

(y(1+x²)) = (sin(x))⁽¹⁰⁰⁾ = sin(x)
Now substituting gives us
y⁽¹⁰⁰⁾+9900y⁽⁹⁸⁾ = sin(0) = 0
Hence, Option 0 is the required answer.

Assume f(x) = y
Rewrite the function as
y(x + 1 ) = ln(x)
Now differentiate both sides up to hundredth derivative in accordance to the Leibniz rule we have

Using the n^th derivative for ln(x+a) as 

we have the right hand side as

Now substituting x = 1 yields

= -(99)!

Explanation:Assume f(x) = y
Rewriting the function as
y² = 1 – x²
Differentiating both sides of the equation up to the third derivative using leibniz rule we have

(1-x²)⁽³⁾=0
Now substituting x = 0 in both the equations and equating them yields
2y ⁽³⁾ y + 6y ⁽²⁾ y⁽¹⁾ = 0.

Explanation:Assume y = f(x) and we also know that tan(x)= 
Rewriting the function as y(cos(x))=sin(x)
Now differentiating on both sides upto n^nt derivative we have
(y(cos(x))⁽ⁿ⁾= 
Now observe that y(0)=tan(0)=0….(1)
Now consider the second derivative at x=0 on both sides

Using (1) and the above equation one can conclude that
y⁽²⁾ = 0
This gives the value of second derivative to be zero
Similarly for any even value of n all the odd derivatives of y in the expression would have sin(x) as their coefficients and as the values of y⁽⁰⁾ and y⁽²⁾ are zero. Every even derivative of the tan(x) function has to be zero.
Thus, we have
y^{(9998879879789776)} = 0.

Explanation: Assume
f(x) = y
Write the given function as
y(x² – x + 1) = cos(x)
Now applying Leibniz rule up to the third derivative we get

Equating both sides we have

Now in the question it is assumed that the y⁽¹⁾=2 and y⁽²⁾=3
Substituting these values in (1) we have

Substituting x = 0 gives
sin(0) = y⁽³⁾ + 9 -12
y⁽³⁾ = 12 – 9 = 3.

Explanation:Rewriting the given function as
y² = x³ + x⁷
Now applying the Leibniz rule up to the third derivative we have
(y²)⁽³⁾=(x³+x⁷)⁽³⁾

Equating both sides and substituting x = 1 we get
y⁽¹⁾ = 0
Now assumed in the question are the values y⁽¹⁾ = 0 and y(2) = 1
We also know y(1) = √2
Putting them in equation (1) we get
2√2 y⁽³⁾ = 3! + 210 = 216
y⁽³⁾ = 54√2.

Expanding sin(x)/x into Taylor series we have

Now Taking the n^th derivative of function using Leibniz rule we have

Now substituting x = 0 we have

Hence, Option  is the right answer.

The key here is to find a separate Taylor expansion for sinh(x) / x which is

Now consider   applying the Leibniz rule for n^th derivative we have
 

Now substituting x = 0 yields