First Principles of Derivatives MCQ Quiz in தமிழ் - Objective Question with Answer for First Principles of Derivatives - இலவச PDF ஐப் பதிவிறக்கவும்

Concept:

1.Derivative of a function with respect to a variable is the rate of change

of a function with respect to the variable.

2. Sum of n terms of an A.P is $\frac{\mathrm{n}}{2}[2\mathrm{a}+(\mathrm{n}-1)\mathrm{d}]$ where n is the number of terms,

a is the first term, and d is the common difference.

Calculation:

f(x) = 1 – x + x2 – x3 ... – x99 + x100

Differentiating the function with respect to x.

⇒ f'(x) = -1 + 2x - 3x² + ..+.. - 99x⁹⁸ + 100 x⁹⁹.

Put x = 1 in the above equation.

⇒ f'(1) = -1 + 2 - 3 + 4 - 5 + ... - 99 + 100.

⇒ f'(1) = (- 1 - 3 - 5 .. - 99) + (2 + 4 + 6 +...+ 100)

Here we have two A.P.

We know sum of n terms of an A.P is $\frac{\mathrm{n}}{2}[2\mathrm{a}+(\mathrm{n}-1)\mathrm{d}]$ where n is the number of terms,

a is the first term, and d is the common difference.

∴ f'(1) = $\frac{50}{2}[2(-1)+(50-1)\times -2]$ + $\frac{50}{2}[2\times 2+(50-1)\times 2]$

⇒ f'(1) = 25 × (- 100) + 25 × 102

⇒ f'(1) = 25 [ - 100 + 102]

⇒ f'(1) = 25 × 2 = 50

Therefore, the required value of f'(1) is 50.

Concept:

$\frac{\mathrm{d}({\mathrm{x}}^{\mathrm{n}})}{\mathrm{d}\mathrm{x}}=\mathrm{n}{\mathrm{x}}^{\mathrm{n}-1}$

$\frac{\mathrm{d}(\mathrm{u}\mathrm{v})}{\mathrm{d}\mathrm{x}}=\mathrm{u}\frac{\mathrm{d}\mathrm{v}}{\mathrm{d}\mathrm{x}}+\mathrm{v}\frac{\mathrm{d}\mathrm{u}}{\mathrm{d}\mathrm{x}}$

$\frac{\mathrm{d}(\sin \mathrm{x})}{\mathrm{d}\mathrm{x}}=\cos \mathrm{x}$

Calculation:

Given: f(x) = x3sinx

f'(x) = $\frac{\mathrm{d}({\mathrm{x}}^3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x})}{\mathrm{d}\mathrm{x}}$

⇒ $\frac{\mathrm{d}({\mathrm{x}}^3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x})}{\mathrm{d}\mathrm{x}}={\mathrm{x}}^3\frac{\mathrm{d}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x})}{\mathrm{d}\mathrm{x}}+\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}\frac{\mathrm{d}{\mathrm{x}}^3}{\mathrm{d}\mathrm{x}}$

= x³ cos x + sin x 3x²

= x²(x cos x + 3sin x)

∴ The required value is x2(x cos x + 3sin x).