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Q17(ii):
Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (ii) by expanding the product to obtain a single polynomial.
Solution :
Step 1: Expansion of the Polynomial Product
Given the function $y = (x^2 - 5x + 8)(x^3 + 7x + 9)$, we apply the distributive property of multiplication over addition [Distributive Law of Algebra]. Each term in the first polynomial is multiplied by each term in the second:
$y = x^2(x^3 + 7x + 9) - 5x(x^3 + 7x + 9) + 8(x^3 + 7x + 9)$
Perform the algebraic expansion:
- $x^2(x^3 + 7x + 9) = x^5 + 7x^3 + 9x^2$
- $-5x(x^3 + 7x + 9) = -5x^4 - 35x^2 - 45x$
- $8(x^3 + 7x + 9) = 8x^3 + 56x + 72$
Step 2: Combining Like Terms
Aggregate the terms by descending order of power [Associative and Commutative Property of Addition]:
$y = x^5 - 5x^4 + (7x^3 + 8x^3) + (9x^2 - 35x^2) + (-45x + 56x) + 72$
$y = x^5 - 5x^4 + 15x^3 - 26x^2 + 11x + 72$
Step 3: Differentiation using Power Rule
To find the derivative $\frac{dy}{dx}$, apply the Sum Rule for differentiation [Sum Rule: $\frac{d}{dx}(u+v) = \frac{du}{dx} + \frac{dv}{dx}$] and the Power Rule [Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$].
$\frac{dy}{dx} = \frac{d}{dx}(x^5) - \frac{d}{dx}(5x^4) + \frac{d}{dx}(15x^3) - \frac{d}{dx}(26x^2) + \frac{d}{dx}(11x) + \frac{d}{dx}(72)$
| Term | Derivative Calculation |
|---|---|
| $x^5$ | $5x^4$ |
| $-5x^4$ | $-5(4x^3) = -20x^3$ |
| $15x^3$ | $15(3x^2) = 45x^2$ |
| $-26x^2$ | $-26(2x) = -52x$ |
| $11x$ | $11(1) = 11$ |
| $72$ | $0$ [Derivative of a constant] |
Step 4: Final Expression Consolidation
Summing the individual derivatives derived in Step 3:
$\frac{dy}{dx} = 5x^4 - 20x^3 + 45x^2 - 52x + 11$
Final Solution: $\frac{dy}{dx} = 5x^4 - 20x^3 + 45x^2 - 52x + 11$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.5
- Q1: Differentiate the functions w.r.t. $x$. $\cos x \cdot \cos 2x \cdot \cos 3x$
- Q2: Differentiate the functions w.r.t. $x$. $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$
- Q3: Differentiate the functions w.r.t. $x$. $(\log x)^{\cos x}$
- Q4: Differentiate the functions w.r.t. $x$. $x^x - 2^{\sin x}$
- Q5: Differentiate the functions w.r.t. $x$. $(x + 3)^2 \cdot (x + 4)^3 \cdot (x + 5)^4$
- Q6: Differentiate the functions w.r.t. $x$. $\left(x + \frac{1}{x}\right)^x + x^{\left(1 + \frac{1}{x}\right)}$
- Q7: Differentiate the functions w.r.t. $x$. $(\log x)^x + x^{\log x}$
- Q8: Differentiate the functions w.r.t. $x$. $(\sin x)^x + \sin^{-1} \sqrt{x}$
- Q9: Differentiate the functions w.r.t. $x$. $x^{\sin x} + (\sin x)^{\cos x}$
- Q10: Differentiate the functions w.r.t. $x$. $x^{x \cos x} + \frac{x^2+1}{x^2-1}$
- Q11: Differentiate the functions w.r.t. $x$. $(x \cos x)^x + (x \sin x)^{\frac{1}{x}}$
- Q12: Find $\frac{dy}{dx}$ of the functions $x^y + y^x = 1$
- Q13: Find $\frac{dy}{dx}$ of the functions $y^x = x^y$
- Q14: Find $\frac{dy}{dx}$ of the functions $(\cos x)^y = (\cos y)^x$
- Q15: Find $\frac{dy}{dx}$ of the functions $xy = e^{(x - y)}$
- Q16: Find the derivative of the function given by $f(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)$ and hence find $f'(1)$.
- Q17(i): Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (i) by using product rule
- Q17(iii): Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (iii) by logarithmic differentiation. Do they all give the same answer?
- Q18: If $u, v$ and $w$ are functions of $x$, then show that $\frac{d}{dx} (u \cdot v \cdot w) = \frac{du}{dx} v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \frac{dw}{dx}$ in two ways - first by repeated application of product rule, second by logarithmic differentiation.
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
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