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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.
Solution :
Method 1: Repeated Application of the Product Rule
Given the product of three functions $f(x) = u \cdot v \cdot w$, we treat $(u \cdot v)$ as a single entity and apply the product rule $[\frac{d}{dx}(f \cdot g) = f'g + fg']$:
Step 1: Let $y = (u \cdot v) \cdot w$. Applying the product rule:
$\frac{dy}{dx} = \frac{d}{dx}(u \cdot v) \cdot w + (u \cdot v) \cdot \frac{d}{dx}(w)$ [By Product Rule]
Step 2: Expand the derivative of the product $(u \cdot v)$:
$\frac{dy}{dx} = \left( \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx} \right) \cdot w + u \cdot v \cdot \frac{dw}{dx}$ [By Product Rule applied to $(u \cdot v)$]
Step 3: Distribute $w$ into the parenthesis:
$\frac{dy}{dx} = \frac{du}{dx} \cdot v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \cdot \frac{dw}{dx}$
Method 2: Logarithmic Differentiation
Step 1: Take the natural logarithm on both sides of the equation $y = u \cdot v \cdot w$:
$\ln(y) = \ln(u \cdot v \cdot w)$
Step 2: Apply the logarithmic property $\ln(abc) = \ln(a) + \ln(b) + \ln(c)$:
$\ln(y) = \ln(u) + \ln(v) + \ln(w)$
Step 3: Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\ln(y)] = \frac{d}{dx}[\ln(u)] + \frac{d}{dx}[\ln(v)] + \frac{d}{dx}[\ln(w)]$
Step 4: Apply the chain rule $\frac{d}{dx}\ln(f) = \frac{1}{f} \cdot f'$:
$\frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} + \frac{1}{v} \cdot \frac{dv}{dx} + \frac{1}{w} \cdot \frac{dw}{dx}$
Step 5: Multiply both sides by $y$:
$\frac{dy}{dx} = y \left( \frac{1}{u} \frac{du}{dx} + \frac{1}{v} \frac{dv}{dx} + \frac{1}{w} \frac{dw}{dx} \right)$
Step 6: Substitute $y = u \cdot v \cdot w$ back into the equation:
$\frac{dy}{dx} = (u \cdot v \cdot w) \left( \frac{1}{u} \frac{du}{dx} + \frac{1}{v} \frac{dv}{dx} + \frac{1}{w} \frac{dw}{dx} \right)$
$\frac{dy}{dx} = (u \cdot v \cdot w \cdot \frac{1}{u} \frac{du}{dx}) + (u \cdot v \cdot w \cdot \frac{1}{v} \frac{dv}{dx}) + (u \cdot v \cdot w \cdot \frac{1}{w} \frac{dw}{dx})$
$\frac{dy}{dx} = \frac{du}{dx} \cdot v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \cdot \frac{dw}{dx}$
Visual Representation of Derivative Summation
Final Solution: $\frac{d}{dx} (u \cdot v \cdot w) = \frac{du}{dx} v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \cdot \frac{dw}{dx}$
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(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.
- 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?
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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