Download App

Differentiation — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferentiation5 Marks
Question
Differentiate the following functions from first principles:
$\text{e}^{\sqrt{2\text{x}}}$
✓

Answer

Let $\text{f(x)}=\text{e}^{\sqrt{2\text{x}}}$
$\Rightarrow\text{f}(\text{x}+\text{h})=\text{e}^{\sqrt{2(\text{x}+\text{h})}}$
$\therefore\frac{\text{d}}{\text{dx}}(\text{f(x)})=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0 }\frac{\text{e}^{\sqrt{2(\text{x}+\text{h})}}-\text{e}^{\sqrt{2\text{x}}}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\text{e}^{\sqrt{2\text{x}}}\frac{\text{e}^{\sqrt{2(\text{x}+\text{h})}-\sqrt{2\text{x}}}-1}{\text{h}}$
$=\text{e}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\Bigg(\frac{\big(\text{e}^{2(\text{x}+\text{h})-\sqrt{2\text{x}}}-1\big)}{\sqrt{2(\text{x}+\text{h})}-\sqrt{2\text{x}}}\Bigg)\bigg(\frac{\sqrt{2(\text{x}+\text{h})}-\sqrt{2\text{x}}}{\text{h}}\bigg)$
$=\text{e}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\frac{\sqrt{2}(\text{x}+\text{h})-\sqrt{2\text{x}}}{\text{h}}\ \Big[\text{Since},\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^\text{h}-1}{\text{h}}=1\Big]$
$=\text{e}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\frac{\sqrt{2(\text{x}+\text{h})}-\sqrt{2\text{x}}}{\text{h}}\times\frac{\sqrt{2(\text{x}+\text{h})}+\sqrt{2\text{x}}}{\sqrt{2(\text{x}+\text{x})}+\sqrt{2\text{x}}}$
$[\text{Rationalizing the numerator]}$
$=\text{de}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\frac{2(\text{x}+\text{h})-2\text{x}}{\text{h}\big(\sqrt{2(\text{x}+\text{h})}+\sqrt{2\text{x}}\big)}$
$=\text{e}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\frac{2\text{x}+2\text{h}-2\text{x}}{\text{h}\big(\sqrt{2}(\text{x}+\text{h})+\sqrt{2\text{x}}\big)}$
$=\text{e}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\frac{2\text{h}}{\text{h}\big(\sqrt{2(\text{x}+\text{h})}+\sqrt{2\text{x}}\big)}$
$=\text{e}^{\sqrt{2\text{x}}}\lim\limits_{\text{h}\rightarrow0}\frac{2\text{h}}{\big(\sqrt{2(\text{x}+\text{h})}+\sqrt{2\text{x}}\big)}$
$=\frac{\text{e}^{\sqrt{2\text{x}}}}{\sqrt{2\text{x}}}$
So,
$\frac{\text{d}}{\text{dx}}\big(\text{e}^{\sqrt{2\text{x}}}\big)=\frac{\text{e}^{2\text{x}}}{\sqrt{2\text{x}}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the maximum and the minimum values, if any, without using derivaives of the following functions:$f(x) = (x - 1)^{2 }+ 2$ on $R.$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x + y}}$
→
An amount of $Rs. 10,000$ is put into three investments at the rate of $10, 12$ and $15\%$ per annum. The combined income is $Rs. 1310$ and the combined income of first and second investment is $Rs. 190$ short of the income from the third. Find the investment in each using matrix method.
→
Find the vector equation (in scalar product form) of the plane containing the line of intersection of the planes x - 3y + 2z - 5 = 0 and 2x - y + 3z - 1 = 0 and passing through (1, -2, 3).
→
m is said to be related to n if m and n are integers and m - n is divisible by 13. Does this define an equivalence relation?
→
A kite is moving horizontally at a height of $151.5$ meters. If the speed of kite is $10m/ s,$ how fast is the string being let out; when the kite is $250m$ away from the boy who is flying the kite? The height of boy is $1.5m$.
→
In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.
→
Differentiate the following functions with respect to x:
$10^{\log\sin\text{x}}$
→
Find graphically, the maximum value of $\text{z = 2x + 5y},$ subject to constraints given below:
$2x + 4y \leq 8$
$3x + y \leq 6$
$x + y \leq 4$
$x\geq 0, y\geq 0$
→
Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}\cos(\text{x}+\text{y})& \sin(\text{x}+\text{y})&\cos2\text{y}\\\sin\text{x}&\cos\text{x}&\sin\text{y}\\-\cos\text{x}&\sin\text{x}&-\cos\text{y} \end{vmatrix}$
→