GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Question 1.
$\frac{e^{x}}{\sin x}$
Solution:
Let y = $\frac{e^{x}}{\sin x}$.

where x ≠ nπ, x ∈ Z.

Question 2.
$e^{\sin ^{-1} x}$
Solution:
Let y = $e^{\sin ^{-1} x}$
Put sin^-1x = t.
∴ y = e^t.

Question 3.
$e^{x^{3}}$
Solution:
Let y = $e^{x^{3}}$. Put x³ = t.
∴ y = e^t and t = x³.
⇒ $\frac { dy }{ dt }$ = e^t and $\frac { dt }{ dx }$ = 3x².
∴ $\frac { dy }{ dx }$ = $\frac { dy }{ dt }$ x $\frac { dt }{ dx }$ = e^t x 3x² = 3x²$e^{x^{3}}$.

Question 4.
sin(tan^-1 e^-x)
Solution:
Let y = sin(tan^-1 e^-x)
Put y = sin s, s = tan^-1 t and t = e^-x

Question 5.
log (cos e^x)
Solution:
Let y = log (cos e^x).
Put y = log s, s = cos t and t = e^x.

Question 6.
e^x + $e^{x^{2}}$ + … + $e^{x^{5}}$
Solution:

Question 7.
$\sqrt{e^{\sqrt{x}}}$, x > 0
Solution:
Let y = $\sqrt{e^{\sqrt{x}}}$.
Put y = $\sqrt{s}$, s = e^t and t = $\sqrt{x}$.
Differentiating, we get

Question 8.
log log x, x > 1
Solution:
Let y = log(log x).
Put y = log t and t = log x.
Differentiating, we get
$\frac { dy }{ dt }$ = $\frac { 1 }{ t }$ and $\frac { dt }{ dx }$ = $\frac { 1 }{ x }$.
∴ $\frac { dy }{ dx }$ = $\frac { dy }{ dt }$ x $\frac { dt }{ dx }$ = $\frac { 1 }{ t }$ x $\frac { 1 }{ x }$ = $\frac { 1 }{ log x }$ . $\frac { 1 }{ x }$
= $\frac { 1 }{ x log x }$, x > 0

Question 9.
$\frac { cos x }{ log x }$, x > 0
Solution:
Let y = $\frac { cos x }{ log x }$.

Question 10.
cos (log x + e^x), x > 0
Solution:
Let y = cos (log x + e^x)
Put y = cos t, t = log x + e^x
Differentiating, we get
$\frac { dy }{ dx }$ = – sin t and $\frac { dt }{ dx }$ = $\frac { 1 }{ x }$ + e^x.
∴ $\frac { dy }{ dx }$ = $\frac { dy }{ dt }$ x $\frac { dt }{ dx }$ = – sin t x ( $\frac { 1 }{ x }$ x e^x).
= – sin (log x + ^x) x ($\frac { 1 }{ x }$ + e^x)
= – $\frac { 1 }{ x }$ (1 + xe^x) sin (log x + e^x).