Differentiate $(x^{2} - 5 x + 8) (x^{3} + 7 x + 9)$ in three ways mentioned below.

(a) By using product rule.

(b) By expanding the product to obtain a single polynomial.

(c) By logarithmic differentiation.

Do they all given the same answer?

Open in App

Solution

Ley $y = (x^{2} - 5 x + 8) (x^{3} + 7 x + 9)$ .,........(i)

(a) Using product rule,

$\frac{d y}{d x} = (x^{2} - 5 x + 8) \frac{d}{d x} (x^{3} + 7 x + 9) + (x^{3} + 7 x + 9) \frac{d}{d x} (x^{2} - 5 x + 8) = (x^{2} - 5 x + 8) (3 x^{2} + 7) + (x^{3} + 7 x + 9) (2 x - 5) = 3 x^{4} - 15 x^{3} + 24 x^{2} + 7 x^{2} - 35 x + 56 + 2 x^{4} + 14 x^{2} + 18 x - 5 x^{3} - 35 x - 45........ (i i) = 5 x^{4} - 20 x^{3} + 45 x^{2} - 52 x + 11$

(b) Writing y as a single polynomial,

$y = (x^{2} - 5 x + 8) (x^{3} + 7 x + 9) = x^{5} - 5 x^{4} + 15 x^{3} - 26 x^{2} + 11 x + 72 D i f f e r e n t i a t i n g b o t h s i d e s w . r . t . x, w e g e t ∴ \frac{d y}{d x} = \frac{d}{d x} (x^{5} - 5 x^{4} + 15 x^{3} - 26 x^{2} + 11 x + 72) = 5 x^{4} - 20 x^{3} + 45 x^{2} - 52 x + 11 . . . . . . (i i i)$

(c) Taking logarithms on both sides of Eq. (i), we get

$l o g y = l o g {(x^{2} - 5 x + 8) (x^{3} + 7 x + 9)} \Rightarrow l o g y = l o g {(x^{2} - 5 x + 8) + l o g (x^{3} + 7 x + 9)} D i f f e r e n t i a t i n g b o t h s i d e s w . r . t . x, w e g e t \frac{1}{y} \frac{d y}{d x} = \frac{1}{x^{2} - 5 x + 8} \frac{d}{d x} (x^{2} - 5 x + 8) + \frac{1}{x^{3} + 7 x + 9} \frac{d}{d x} (x^{3} + 7 x + 9) (U s i n g c h a i n r u l e) \frac{1}{y} \frac{d y}{d x} = \frac{2 x - 5}{x^{2} + 5 x + 8} + \frac{3 x^{2} + 7}{x^{3} + 7 x + 9} = (x^{2} - 5 x + 8) (x^{3} + 7 x + 9) (\frac{2 x - 5}{x^{2} - 5 x + 8} + \frac{3 x^{2} + 7}{x^{3} + 7 x + 9}) = (2 x - 5) (x^{3} + 7 x + 9) + (3 x^{2} + 7) (x^{2} - 5 x + 8) = 2 x^{4} + 14 x^{2} + 18 x - 5 x^{3} - 35 x - 45 + 3 x^{4} - 15 x^{3} + 24 x^{2} + 7 x^{2} - 35 x + 56 = 5 x^{4} - 20 x^{3} + 45 x^{2} - 52 x + 11 . . . . . . . . . (i v)$

From Eqs. (ii), (iii) and (iv), we find that result is same in every case.

Suggest Corrections

Similar questions

Differentiate in three ways mentioned below

(i) By using product rule.

(ii) By expanding the product to obtain a single polynomial.

(iii By logarithmic differentiation.

Do they all give the same answer?

(x^{2} - 5 x + 8) (x^{3} + 7 x + 9)

\begin{matrix} Differentiate (x^{2} - 5 x + 8) (x^{3} + 7 x + 9) in ways mentioned below: (i) by using product rule \end{matrix}

Join BYJU'S Learning Program