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73. Let y = (x + 3)² (x + 4)³ (x + 5)⁴.

Taking log_e on both sides,

log y = log (x + 3)² + log (x + 4)³ + log (x + 5)⁴

= 2 log (x + 3) + 3 (log (x + 4) + 4 log (x + 5).

So,

Q $\frac{d}{dx}$ log y = $\frac{d}{dx}$  [2 log (x + 3) + 3 log (x + 4) + 4 log (x +5)]

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{2}{x+3}+\frac{3}{x+4}+\frac{4}{x+5.}$

$\Rightarrow \frac{dy}{dx}=y\left[\frac{2}{x+3}+\frac{3}{x+4}+\frac{4}{x+5}\right]$

$\Rightarrow \frac{dy}{dx}$ = (x + 3)² (x + 4)³ (x + 5)⁴${\left[\frac{2}{x+3}+\frac{3}{x+4}+\frac{4}{x+5}\right]}_{.}$

To get admission to the CHARUSAT BPharm programme, candidates must ensure that they meet the requisite eligibility criteria. The selection of candidates for the BPharm programme is based on the scores obtained in the accepted entrance exam. Those who are selected are required to present the necessary documents for verification.

Yes, there are many government BA in Psychology colleges in Delhi NCR. Some of them are mentioned below along with their tuition fees:

Disclaimer: This information is sourced from the official website and may vary.

72. Let y = x^x - 2 sin x

Putting u = x^x and v = 2 sin x.

So, y = u - v

= $\frac{dy}{dx}=\frac{dy}{dx}-\frac{dv}{dx}$ ____ (i)

As u = x^x

Log u = x log x.

So,$\frac{d}{dx}$ log u = $\frac{d}{dx}$ x log x.

= $\frac{1}{4}\frac{dy}{dx}=\frac{xdlogx}{dx}+logx\frac{dx}{dx}$

x´ $\frac{1}{x}$ + log x.

= 1 + log x

= $\frac{dy}{dx}$= 4 [1 + log x] = x^x [1 + log x].

And v = 2sin x Log v = sin x log 2.

$\frac{d}{dx}(logv)=\frac{d}{dx}$(sin x log 2)

$\Rightarrow \frac{1}{v}\frac{dv}{dx}=sinx\frac{d}{dx}$ sin x $\frac{d}{dx}$ log 2 + log 2 $\frac{dsinx}{dx}$ = log 2. cos x.

$\Rightarrow \frac{dv}{dx}$ = v log2 cos x.

$\Rightarrow \frac{dv}{dx}$ = v log 2 cos x

= 2 ^{sin x} log². cos x.

Q Eqⁿ (i) becomes,    = x^x (1 + log x) - 2 ^{sin x} cos x log 2.

Sambalpur University has more than 20 affiliated colleges, amounting to an intake of 31,571 seats for BA courses. Among all the colleges/institutes, Government College Bhawanipatna has the highest intake with 464 seats. In order to get a seat, students must first apply through the admission portal available on the admission portal of Sambalpur University.

The institute has not released its course wise data yet. However, NIRF Report for 2025 is out. According to the NIRF report 16 PG 2-year students were placed during  SCSVMV placements. Refer to the table below to know more:

Note: The data provided above is taken from NIRF report 2024

14. Kindly go through the solution

71. Let y = (log x) cos x

Taking loge on both sides,

Log y = cos x [log (log x)]

Differentiating w r t ‘x’ we get,

$\frac{1}{y}\frac{dy}{dx}=cosx\frac{d}{dx}$ log (log x) +log (log x) $\frac{dcosx}{dx}$

$=cosx\times \frac{1}{logx}\times \frac{d(logx)}{dx}$ + log (log x) (- sin x)

$=\frac{cogx}{xlogx}$ - sin x log (log x)

= $\frac{dy}{dx}=y{\left[\frac{cosx}{xlog2}-sinx·log(logx)\right]}_{.}$

$\Rightarrow \frac{dy}{dx}={(logx)}^{cosx}\left[\frac{cosx}{xlogx}-sinx·log(logx)\right]$

The PGDAV College provides good placement opportunities to its graduating students. So far as of 2024, the highest package offered stood at INR 23 LPA with 49 companies that participated in the placements. Check out the table for more placement details:

Note- The college has not released the placement report for season 2023 as of yet. 

The institution has not released the course-wise placement packages as of yet. In 2024, the highest package offered stood at INR 23 LPA. Meanwhile the placements for PGDAV College recorded in 2021 and 2022 are listed below:

Note- The college has not released the placement report for season 2023 as of yet. 

70. Kindly go through the solution

Putting value of y from the above we get,

Yes, PGDAV College has an impressive placement record. As per the annual report, 2024 the highest package offered stood at INR 23 LPA. The table below provides the placement data for the 2021 and 2022 placement as per the placement brochure:

Note- The college has not released the placement report for season 2023 as of yet. 

As per official sources, Sambalpur University offers BA courses via more than 20 colleges and institutes. Thus, the university offers a total of 31,571 seats for BA admissions. Moreover, the allocation of these seats is done based on the last qualifying exam scores. However, students must note that the selection criteria may vary depending on the choice of institute.

69. Let y = cos x cos 2x cos 3x _____ (i)

Taking log_e on bolk sides.

logy = log (cos x) + log (cos 2x + log (cos 3x)

= log (cos x) + log (cos 2x) + log (cos 3x)

Differentiating w r t ‘x’

$\frac{1}{y}\frac{dy}{dx}=\frac{1}{cosx}\frac{d}{dx}cosx+\frac{1}{cos2x}\frac{d}{dx}cos2x+\frac{1}{cos3x}\frac{d}{dx}cos3x.$

$=\frac{(-sinx)}{cosx}+\frac{(-sin2x)}{cos2x}\frac{d}{dx}(2x)+\frac{(-sin3x)}{cos3x}\frac{d(3x)}{dx}$

= - tan x-2 tan 2x- 3 tan 3x.

$\Rightarrow \frac{dy}{dx}$ = y [- tan x- 2 tan 2x- 3 tan 3x]

Putting value of y from (i) we get,

$=\frac{dy}{dx}$= - cos x cos 2x cos 3x [tan x + 2 tan 2x + 3 tan 3x]

GDAV College provides good placements to its graduating students. In 2024, a total of 49 companies participated and offered a highest package of INR 23 LPA. Check out the table below to know the key highlights of the PGDAV College placement in 2021-2022:

Note- The college has not released the placement report for season 2023 as of yet. 

The institute has not released its course wise data yet. However, NIRF Report for 2025 is out. According to the NIRF report 156 PG 2-year students were placed during Manonmaniam Sundaranar University placements. Refer to the table below to know more:

Note: The data provided above is taken from NIRF report 2024