Question

if f(x)=(a^(x))/(x^(a)) then f'(a)=​

Answer 1

Answer:

Correct option is

B

loga−a

According to question,

f′(x)=x2aaxlogaxa−xa+1ax

⇒f′(a)=a2aaaloga.aa−aa+1aa

Hence,

 ⇒f′(a)=loga−a

Answer 2

Answer:

f'(a) = loga - 1

Step-by-step explanation:

$\sf \bf :\longmapsto f(x) = \dfrac{a^x}{x^a}$

Taking log both sides,

$\sf \bf :\longmapsto log (f(x)) = log\left(\dfrac{a^x}{x^a}\right)$

$\sf \bf :\longmapsto log(f(x)) = loga^x - logx^a$

$\sf \bf :\longmapsto log(f(x)) = xloga - alogx$

Differentiating to find f'(x),

Use the property,

$\boxed{\boxed{ \bf \tt \dfrac{d}{dx}logx = \dfrac{1}{x}}}$

Thus,

$\sf \bf :\longmapsto \dfrac{1}{f(x)} f'(x) = loga - \dfrac{a}{x}$

$\sf \bf :\longmapsto f'(x) = \dfrac{a^x}{x^a} \left(loga - \dfrac{a}{x}\right)$

$\sf \bf :\longmapsto f'(a) = \dfrac{a^a}{a^a} \left(loga - \dfrac{a}{a}\right)$

$\sf \bf :\longmapsto f'(a) = loga - 1$

Hence,

$\boxed{\boxed{\sf \bf f'(a) = loga - 1}}$

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