R.D. Sharma solutions for माठेमटिक्स १ [इंग्रजी] इयत्ता ११ chapter 28 - Derivatives [Latest edition]

Find the derivative of f (x) = 3x at x = 2 

Find the derivative of f (x) = x² − 2 at x = 10

Find the derivative of f (x) = 99x at x = 100 

Find the derivative of f (x) = cos x at x = 0

Find the derivative of f (x) = tan x at x = 0 

Find the derivative of the following function at the indicated point: 

 sin x at x =\[\frac{\pi}{2}\]

Find the derivative of the following function at the indicated point:

Find the derivative of the following function at the indicated point:

2 cos x at x =\[\frac{\pi}{2}\] 

Find the derivative of the following function at the indicated point: 

 sin 2x at x =\[\frac{\pi}{2}\]

\[\frac{2}{x}\]

\[\frac{1}{\sqrt{x}}\]

\[\frac{1}{x^3}\]

\[\frac{x^2 + 1}{x}\]

\[\frac{x^2 - 1}{x}\]

\[\frac{x + 1}{x + 2}\]

\[\frac{x + 2}{3x + 5}\]

k xⁿ

\[\frac{1}{\sqrt{3 - x}}\]

 x² + x + 3

(x + 2)³

 (x² + 1) (x − 5)

 (x² + 1) (x − 5)

\[\sqrt{2 x^2 + 1}\]

\[\frac{2x + 3}{x - 2}\] 

Differentiate each of the following from first principle:

e^−x

Differentiate  of the following from first principle:

e^3x

Differentiate  of the following from first principle:

 e^ax + b

x e^x

Differentiate  of the following from first principle: 

− x

Differentiate of the following from first principle:

(−x)⁻¹

Differentiate  of the following from first principle:

sin (x + 1)

Differentiate  of the following from first principle:

\[\cos\left( x - \frac{\pi}{8} \right)\]

Differentiate  of the following from first principle:

 x sin x

Differentiate of the following from first principle:

 x cos x

Differentiate  of the following from first principle:

sin (2x − 3)

Differentiate each of the following from first principle:

\[\sqrt{\sin 2x}\] 

Differentiate each of the following from first principle:

\[\frac{\sin x}{x}\]

Differentiate each of the following from first principle: 

\[\frac{\cos x}{x}\]

Differentiate each of the following from first principle:

 x² sin x

Differentiate each of the following from first principle:

\[\sqrt{\sin (3x + 1)}\]

Differentiate each of the following from first principle: 

sin x + cos x

Differentiate each of the following from first principle:

x² e^x 

Differentiate each of the following from first principle: 

\[e^{x^2 + 1}\]

Differentiate each  of the following from first principle:

\[e^\sqrt{2x}\]

Differentiate each of the following from first principle:

\[e^\sqrt{ax + b}\]

Differentiate each of the following from first principle:

\[a^\sqrt{x}\]

Differentiate each of the following from first principle:

\[3^{x^2}\]

tan² x 

tan (2x + 1) 

 tan 2x 

\[\sqrt{\tan x}\]

\[\sin \sqrt{2x}\]

\[\cos \sqrt{x}\]

\[\tan \sqrt{x}\]

\[\tan \sqrt{x}\] 

x⁴ − 2 sin x + 3 cos x

3^x + x³ + 3³

\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]

e^x log a + e^a long x + e^a log a

(2x² + 1) (3x + 2) 

 log₃ x + 3 log_e x + 2 tan x

\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\] 

\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\] 

\[\frac{2 x^2 + 3x + 4}{x}\] 

\[\frac{( x^3 + 1)(x - 2)}{x^2}\] 

\[\frac{a \cos x + b \sin x + c}{\sin x}\]

2 sec x + 3 cot x − 4 tan x

a₀ xⁿ + a₁ xⁿ⁻¹ + a₂ xⁿ⁻² + ... + a_n−1 x + a_n. 

\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\] 

\[\frac{(x + 5)(2 x^2 - 1)}{x}\]

\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\] 

cos (x + a)

\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]

\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]

Find the slope of the tangent to the curve f (x) = 2x⁶ + x⁴ − 1 at x = 1.

\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]  

Find the rate at which the function f (x) = x⁴ − 2x³ + 3x² + x + 5 changes with respect to x.

\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\] 

If for f (x) = λ x² + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ. 

For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]

x³ sin x 

x³ e^x 

x² e^x log x 

xⁿ tan x 

xⁿ log_a x 

(x³ + x² + 1) sin x 

sin x cos x

\[\frac{2^x \cot x}{\sqrt{x}}\] 

x² sin x log x 

x⁵ e^x + x⁶ log x 

(x sin x + cos x) (x cos x − sin x) 

(x sin x + cos x ) (e^x + x² log x) 

(1 − 2 tan x) (5 + 4 sin x)

(1 +x²) cos x

sin² x 

log_x² x

\[e^x \log \sqrt{x} \tan x\] 

x³ e^x cos x 

\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\] 

x⁴ (5 sin x − 3 cos x)

(2x² − 3) sin x 

x⁵ (3 − 6x⁻⁹) 

x⁻⁴ (3 − 4x⁻⁵)

x⁻³ (5 + 3x) 

Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same. 

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same. 

 (3x² + 2)²

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(x + 2) (x + 3)

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)

(ax + b) (a + d)²

(ax + b)ⁿ (cx + d)ⁿ 

\[\frac{x^2 + 1}{x + 1}\] 

\[\frac{2x - 1}{x^2 + 1}\] 

\[\frac{x + e^x}{1 + \log x}\] 

\[\frac{e^x - \tan x}{\cot x - x^n}\] 

\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\] 

\[\frac{x}{1 + \tan x}\] 

\[\frac{1}{a x^2 + bx + c}\] 

\[\frac{e^x}{1 + x^2}\] 

\[\frac{e^x + \sin x}{1 + \log x}\] 

\[\frac{x \tan x}{\sec x + \tan x}\]

\[\frac{x \sin x}{1 + \cos x}\]

\[\frac{2^x \cot x}{\sqrt{x}}\] 

\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]

\[\frac{x^2 - x + 1}{x^2 + x + 1}\] 

\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\] 

\[\frac{a + \sin x}{1 + a \sin x}\] 

\[\frac{{10}^x}{\sin x}\] 

\[\frac{1 + 3^x}{1 - 3^x}\]

\[\frac{3^x}{x + \tan x}\] 

\[\frac{1 + \log x}{1 - \log x}\] 

\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]

\[\frac{x}{1 + \tan x}\] 

\[\frac{a + b \sin x}{c + d \cos x}\] 

\[\frac{p x^2 + qx + r}{ax + b}\]

\[\frac{\sec x - 1}{\sec x + 1}\] 

\[\frac{x^5 - \cos x}{\sin x}\] 

\[\frac{x + \cos x}{\tan x}\] 

\[\frac{x}{\sin^n x}\]

\[\frac{ax + b}{p x^2 + qx + r}\] 

\[\frac{1}{a x^2 + bx + c}\] 

Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\] 

Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]

If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\] 

If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]

Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]

Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]

If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\] 

Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]

If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\] 

Write the derivative of f (x) = 3 |2 + x| at x = −3. 

If |x| < 1 and y = 1 + x + x² + x³ + ..., then write the value of \[\frac{dy}{dx}\] 

If f (x) =  \[\log_{x_2}\]write the value of f' (x). 

Mark the correct alternative in of the following:

Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]

Mark the correct alternative in of the following: 

If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]

Mark the correct alternative in of the following:

If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\] 

Mark the correct alternative in  of the following:

If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\] 

Mark the correct alternative in  of the following: 

If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\] 

Mark the correct alternative in of the following:

If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is

Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\]  then \[f'\left( 1 \right)\] is equal to 

Mark the correct alternative in  of the following:
If\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to 

Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is 

Mark the correct alternative in  of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is 

Mark the correct alternative in each of the following: 

If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\] 

Mark the correct alternative in of the following: 

If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]