मराठी

Differentiate Each of the Following from First Principle:\[A^\Sqrt{X}\]

Advertisements
Advertisements

प्रश्न

Differentiate each of the following from first principle:

\[a^\sqrt{x}\]

Advertisements

उत्तर

\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( a^\sqrt{x} \right) = \lim_{h \to 0} \frac{a^\sqrt{x + h} - a^\sqrt{x}}{h}\]
\[ = \lim_{h \to 0} \frac{a^\sqrt{x} \left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( x + h \right) - \left( x \right)}\]
\[ = a^\sqrt{x} \lim_{h \to 0} \frac{\left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( \sqrt{x + h} \right)^2 - \left( \sqrt{x} \right)^2}\]
\[ = a^\sqrt{x} \lim_{h \to 0} \frac{\left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( \sqrt{x + h} - \sqrt{x} \right)\left( \sqrt{x + h} + \sqrt{x} \right)}\]
\[ = a^\sqrt{x} \lim_{h \to 0} \frac{\left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( \sqrt{x + h} - \sqrt{x} \right)} \lim_{h \to 0} \frac{1}{\left( \sqrt{x + h} + \sqrt{x} \right)}\]
\[ = a^\sqrt{x} \log_e a \frac{1}{2\sqrt{x}}\]
\[ = \frac{1}{2\sqrt{x}} a^\sqrt{x} \log_e a\]
\[\]

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 30: Derivatives - Exercise 30.2 [पृष्ठ २६]

APPEARS IN

आर.डी. शर्मा Mathematics [English] Class 11
पाठ 30 Derivatives
Exercise 30.2 | Q 3.11 | पृष्ठ २६

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

For the function

f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`

Prove that f'(1) = 100 f'(0)


Find the derivative of x–3 (5 + 3x).


Find the derivative of x5 (3 – 6x–9).


Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

`(px+ q) (r/s + s)`


Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

cosec x cot x


Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

`cos x/(1 + sin x)`


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


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


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


k xn


 x2 + x + 3


x ex


Differentiate  of the following from first principle:

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


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:

\[3^{x^2}\]


tan (2x + 1) 


\[\tan \sqrt{x}\]


x4 − 2 sin x + 3 cos x


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


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


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


x3 e


x2 ex log 


sin x cos x


x3 ex cos 


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. 


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


\[\frac{1 + \log x}{1 - \log x}\] 


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


\[\frac{1}{a x^2 + bx + c}\] 


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 \[\frac{d}{dx}\left( f (x) \right)\]


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


Mark the correct alternative in of the following:

If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] 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 


`(a + b sin x)/(c + d cos x)`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×