First Principles of Derivatives MCQ Quiz in मल्याळम - Objective Question with Answer for First Principles of Derivatives - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

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 \(\rm\frac{n}{2}\left[2a+(n-1)d\right]\) 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 \(\rm\frac{n}{2}\left[2a+(n-1)d\right]\) where n is the number of terms,

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

∴ f'(1) = \(\frac{50}{2}\left[2(-1)+(50-1)× -2\right]\) + \(\frac{50}{2}\left[2×2+(50-1)× 2\right]\)

⇒ 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:

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

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

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

Calculation:

Given: f(x) = x3sinx

f'(x) = \(\rm \frac{\mathrm{d} (x^{3}sinx)}{\mathrm{d} x}\)

⇒ \(\rm \frac{\mathrm{d} (x^{3}sinx)}{\mathrm{d} x} = x^{3}\frac{\mathrm{d} (sinx)}{\mathrm{d} x} + sinx \frac{\mathrm{d} x^{3}}{\mathrm{d} x}\)

= x³ cos x + sin x 3x²

= x²(x cos x + 3sin x)

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