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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability3 Marks
Question
Differentiate w.r.t. $x$ the function in $x^x + x^a + a^x + a^a ,$ for some fixed $a > 0$ and $x > 0.$
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Answer

Let $y = x^x + x^a + a^x + a^a,$ for some fixed a > 0 and $x > 0$
And let $x^x = u, x^a = v a^x = w$ and $a^a = s$
Then $y = u + v + w + s$
$\therefore$ $\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}+\frac{d w}{d x}+\frac{d s}{d x} ......(i)$
Now,
$u = x^x$
Taking logarithm both sides, we get
$\log u = \log x^x$​​​​​​​
$\Rightarrow \log u = x \log x$
Differentiating both sides w.r.t. $x$
$\frac{1}{\mathrm{u}} \frac{\mathrm{du}}{\mathrm{dx}}=\log \mathrm{x} \times \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x})+\mathrm{x} \times \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})$
$\Rightarrow \frac{d u}{d x}=u\left[\log x+x \times \frac{1}{x}\right]$
$\Rightarrow \frac{d u}{d x}=x^{x}[\log x+1]=x^{x}(1+\log x) ......(ii)$
$v = x^a$​​​​​​​
Differentiating both sides with respect to $x$
$\frac{d v}{d x}=\frac{d}{d x}\left(x^{a}\right)$
$\Rightarrow \frac{d v}{d x}=a x^{a-1} .....(iii)$
$w = a^x$​​​​​​​
Taking logarithm both sides
$\log w = \log a^x$
$\log w = x \log a$
Differentiating both sides with respect to $x$
$\frac{1}{w} \frac{d w}{d x}=\log a \times \frac{d}{d x}(x)$
$\Rightarrow \frac{d w}{d x}=w \log a$
$\Rightarrow \frac{d w}{d x}=a^{x} \log a ......(iv)$
$s = a^a$​​​​​​​
Differentiating both sides with respect to $x$
$\frac{\mathrm{ds}}{\mathrm{dx}}=0 .....(v)$
Putting $(ii), (iii), (iv)$ and $(v)$ in $(i)$
$\frac{d y}{d x}=x^{x}(1+\log x)+a x^{a-1}+a^{x} \log a+0$
$\therefore \frac{d y}{d x}=x^{x}(1+\log x)+a x^{a-1}+a^{x} \log a$

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