AP Inter 2nd Year Maths Exercise 5d Solutions

Referring to the AP Inter 2nd Year Maths Study Material Chapter 5 Continuity and Differentiability Exercise 5d Solutions makes it easier to understand complex problems.

AP Inter 2nd Year Maths Continuity and Differentiability Solutions Exercise 5d

Question 1.
Differentiate \(\frac{e^x}{\sin x}\) w.r.t. x
Solution:

Question 2.
Differentiate e^sin-1x w.r.t. x
Solution:
Let y = e^sin-1x
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = e^sin-1x \(\frac{\mathrm{d}}{\mathrm{dx}}\)sin^-1x [∵ \(\frac{\mathrm{d}}{\mathrm{dx}}\) e^f(x) = e^f(x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)f(x)]
= e^sin-1x \(\frac{1}{\sqrt{1-x^2}}\), x ∈ (-1, 1)

Question 3.
Differentiate e^x3 w.r.t. x.
Solution:
Let y = e^x3 = e^(x3)
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = e^(x3) \(\frac{\mathrm{d}}{\mathrm{dx}}\)x³ [∵ \(\frac{\mathrm{d}}{\mathrm{dx}}\) e^f(x) = e^f(x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)f(x)]
= e^(x3)3x² = 3x²e^(x3)

Question 4.
Differentiate sin(tan^-1e^-x) w.r.t. x:
Solution:
Let y = sin(tan^-1e^-x)

Question 5.
Differentiate log(cos e^x) w.r.t. x.
Solution:
Let y = log(cos e^x)
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{\cos e^x} \frac{d}{d x}\) (cos e^x)
= \(\frac{1}{\cos \mathrm{e}^{\mathrm{x}}}\)(-sin e^x) \(\frac{\mathrm{d}}{\mathrm{dx}}\) e^x
= -(tan e^x)e^x = -e^x (tan e^x)

Question 6.
Differentiate e^x + e^x2+ ……….. + e^x5 w.r.t. x.
Solution:
Let y = e^x + e^x2+ e^x3 + e^x4 + e^x5
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\mathrm{d}}{\mathrm{dx}}\)e^x + \(\frac{\mathrm{d}}{\mathrm{dx}}\)e^x2+ \(\frac{\mathrm{d}}{\mathrm{dx}}\) e^x3 + \(\frac{\mathrm{d}}{\mathrm{dx}}\) e^x4 + ……….. + \(\frac{\mathrm{d}}{\mathrm{dx}}\)e^x5
= e^x + e^x2\(\frac{\mathrm{d}}{\mathrm{dx}}\) x² + e^x3 \(\frac{\mathrm{d}}{\mathrm{dx}}\) x³ + e^x4\(\frac{\mathrm{d}}{\mathrm{dx}}\) x⁴ + ……….. + e^x5 \(\frac{\mathrm{d}}{\mathrm{dx}}\) x⁵ [∵ \(\frac{\mathrm{d}}{\mathrm{dx}}\) e^f(x) = e^f(x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)f(x)]
= e^x + e^x2 2x + e^x3 3x² + e^x4 4x³ + e^x5 5x⁴
= e^x + 2x e^x2+ 3x² e^x3 + 4x³ e^x4 + 5x⁴ e^x5

Question 7.
Differentiate \(\sqrt{e^{\sqrt{x}}}\), x > 0 w.r.t. x.
Solution:
Let y = \(\sqrt{e^{\sqrt{x}}}\) = \(\left(e^{\sqrt{x}}\right)^{1 / 2}\)
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{2}\left(e^{\sqrt{x}}\right)^{-1 / 2} \frac{d}{d x} e^{\sqrt{x}}\)
= \(\frac{1}{2 \sqrt{e^{\sqrt{x}}}} e^{\sqrt{x}} \frac{d}{d x} \sqrt{x}\) = \(\frac{1}{2 \sqrt{e^{\sqrt{x}}}} e^{\sqrt{x}} \frac{1}{2 \sqrt{x}}\)
= \(\frac{e^{\sqrt{x}}}{4 \sqrt{x} \sqrt{e^{\sqrt{x}}}}\)=\(\frac{e^{\sqrt{x}}}{4 \sqrt{x e^{\sqrt{x}}}}\)

Question 8.
Differentiate log (log x), x > 1 w.r.t. x
Solution:
Let y = log (log x)
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{\log x} \frac{d}{d x}(\log x)\) [∵ \(\frac{\mathrm{d}}{\mathrm{dx}}\) logf(x) = \(\frac{1}{f(x)} \frac{d}{d x}\) f(x)]
= \(\frac{1}{\log x} \frac{d}{d x}(\log x)\)
= \(\frac{1}{x \log x}\)

Question 9.
Differentiate \(\frac{\cos x}{\log x}\), x > 0 w.r.t. x
Solution:

Question 10.
Differentiate cos(logx + e^x), x > 0 w.r.t x.
Solution:
Let y = cos(logx + e^x)
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = -sin(log x + e^x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(log x + e^x)
= -sin(log x + e^x) . (\(\frac{1}{x}\) + e^x) = -(\(\frac{1}{x}\) + e^x)sin(log x + e^x)