Exercise 5.7 - Chapter 5 - Differential Calculus - 11th Business Maths Guide Samacheer Kalvi Solutions

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Exercise 5.7

 Text Book Back Questions and Answers

Question 1.
Differentiate the following with respect to x.
(i) x^{sin x}
(ii) (sin x)^x
(iii) (sin x)^{tan x}

Solution:
(i) Let y = x^{sin x}
Taking logarithm on both sides we get,
log y = log(x^{sin x})
log y = sin x log x
Differentiating with respect to x,

(ii) Let y = (sin x)^x
Taking logarithm on both sides we get,
log y = x log(sin x)
Differentiating with respect to x,

(iii) Let y = (sin x)^{tan x}
Taking logarithm on both sides we get,
log y = tan x log(sin x)
Differentiating with respect to x,

Taking logarithm on both sides we get,
log y = 1/2 {[log(x – 1) + log(x – 2)] – [(log(x – 3) + log(x² + x + 1)]}
log y = 1/2 [log(x – 1) + log(x – 2) – log(x – 3) – log(x² + x + 1)]
Differentiating with respect to x,

Question 2.
If x^m . yⁿ = (x + y)^m+n, then show that 

Solution:
x^m . yⁿ = (x + y)^m+n
Taking logarithm on both sides we get,
m log x + n log y = (m + n) log(x + y)
Differentiating with respect to x,

Hence proved.