Exercise 5.5 - Chapter 5 - Differential Calculus - 11th Business Maths Guide Samacheer Kalvi Solutions

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Exercise 5.5

Text Book Back Questions and Answers

Question 1.
Differentiate the following with respect to x.
(i) 3x⁴ – 2x³ + x + 8

Solution:

(i) Let y = 3x⁴ – 2x³ + x + 8

(v) Let y = x³ e^x
[∵ We have product of two functions, so use product rule]
= x³ e^x + e^x (3x²)
= e^x (x³ + 3x²)
= x² e^x (x + 3)

(vi) Let y = (x² – 3x + 2) (x + 1)
y = x³ – 3x² + 2x + x² – 3x + 2
y = x³ – 2x² – x + 2

(vii) Let y = x⁴ – 3 sin x + cos x

= 4x³ – 3 cos x – sin x

Question 2.
Differentiate the following with respect to x.

Solution:

Question 3.
Differentiate the following with respect to x.
(i) x sin x
(ii) e^x sin x
(iii) e^x (x + log x)
(iv) sin x cos x
(v) x³ e^x
Solution:
(i) Let y = x sin x

= x cos x + sin x . 1
= x cos x + sin x

(ii) Let y = e^x sin x

= e^x cos x + sin x e^x
= e^x (cos x + sin x)

(iii) Let y = e^x (x + log x)

Question 4.
Differentiate the following with respect to x.
(i) sin² x
(ii) cos² x
(iii) cos³ x

Solution:
For the following problems chain rule to be used: