COMPASSLearning

Talk to us Register Login

JEE Main · 2022 · 27 Jun 2022 · Shift 2 · Differentiation

If

y (x) = (x^{x})^{x}, x > 0

then

\frac{d ^{2} x}{dy ^{2}} + 20

at

x = 1

is equal to:

Answer: 16

Solution

Read as

y = (x^{x})^{x} = x^{x^{2}}

, take logarithms to get

ln y = x^{2} ln x

. Differentiating,

\frac{y ^{'}}{y} = 2 x ln x + x

, so

y^{'} = y (2 x ln x + x)

, and at

x = 1

, where

y = 1

, this is

y^{'} = 1 \cdot (0 + 1) = 1

. Differentiating once more,

y^{''} = y^{'} (2 x ln x + x) + y (2 ln x + 3)

, which at

x = 1

equals

1 \cdot 1 + 1 \cdot 3 = 4

. For the inverse function the second derivative obeys

\frac{d ^{2} x}{d y ^{2}} = - \frac{y ^{''}}{( y ^{'} ) ^{3}} = - \frac{4}{1 ^{3}} = - 4

, so

\frac{d ^{2} x}{d y ^{2}} + 20 = - 4 + 20 = 16

.

JEE Main · 2022 · 27 Jun 2022 · Shift 2 · Integer answer

Practise more: all JEE Main PYQs on Differentiation →

More JEE Main questions on Differentiation

Let $f (x) = x^{3} + x^{2} f^{'} (1) + 2 x f^{''} (2) + f^{'''} (3)$ , $x \in R$ . Then the value of $f^{'} (5)$ is :
JEE Main 2026View solution →
If $y (x) = sin x 271 cos x 281 sin x + cos x + 1 271$ , $x \in R$ , then $\frac{d ^{2} y}{d x ^{2}} + y$ is equal to
JEE Main 2025View solution →
Let $f (x) = x^{5} + 2 e^{x /4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g \circ f) (x) = x$ for all $x \in R$ . Then the value of $8 g^{'} (2)$ is
JEE Main 2024View solution →
Let $f (x) = x^{5} + 2 x^{3} + 3 x + 1, x \in R$ , and $g (x)$ be a function such that $f (g (x)) = x$ for all $x \in R$ . Then $\frac{g ( 7 )}{g ^{'} ( 7 )}$ is equal to:
JEE Main 2024View solution →
If $y (θ) = \frac{2 c o s θ + c o s 2 θ}{c o s 3 θ + 4 c o s 2 θ + 5 c o s θ + 2}$ , then at $θ = \frac{π}{2}$ , $y^{''} + y^{'} + y$ is equal to
JEE Main 2024View solution →
Suppose for a function $h, h (0) = 0, h (1) = 1$ and $h^{'} (0) = h^{'} (1) = 2$ . If $g (x) = h (e^{x}) e^{h (x)}$ , then $g^{'} (0)$ is equal to
JEE Main 2024View solution →