Download App

Derivatives — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSDerivatives5 Marks
Question
Differentiate the following from first principle

$\text{a}^{\sqrt{\text{x}}}$

✓

Answer

$\text{f}(\text{x})=\text{a}^\sqrt{\text{x}}=\text{e}^{\sqrt{\text{x}}\log\text{a}}$

$\text{f}'(\text{x})=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$

$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^{\sqrt{\text{x}+\text{h}}\log\text{a}}-\text{e}^{\sqrt{\text{x}}\log\text{a}-1}}{\text{h}}$

$=\lim_\limits{\text{h}\rightarrow0}\text{e}^{\sqrt{\text{x}}\log\text{a}}\frac{\text{e}^{\sqrt{\text{x}+\text{h}}\log\text{a}-\sqrt{\text{x}}\log\text{a}}-1}{\text{h}}$

$=\lim_\limits{\text{h}\rightarrow0}\text{e}^{\sqrt{\text{x}}\log\text{a}}\frac{\text{e}^{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}}-1}{\text{h}}$

Multiply numerator and denominator by $(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}$

$\text{f}(\text{x})=\lim_\limits{\text{h}\rightarrow0}\text{e}^{\sqrt{\text{x}}\log\text{a}}\frac{\text{e}^{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}}-1}{\text{h}(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}}(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}$

$=\text{e}^{\sqrt{\text{x}}\log\text{a}}\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}}-1}{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})\log\text{a}}\lim_\limits{\text{h}\rightarrow0}\log\text{a}\frac{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})}{\text{h}}$

$=\text{e}^{\sqrt{\text{x}}\log\text{a}}\lim_\limits{\text{h}\rightarrow0}\log\text{a}\frac{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})}{\text{h}}$

Multiply numerator and denominator by $(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})$

$\text{f}(\text{x})=\text{e}^{\sqrt{\text{x}}\log\text{a}}\lim_\limits{\text{h}\rightarrow0}\log\text{a}\frac{(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})}{\text{h}(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})}(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})$

$=\text{e}^{\sqrt{\text{x}}\log\text{a}}\lim_\limits{\text{h}\rightarrow0}\log\text{a}\frac{\text{h}}{\text{h}(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}})}$

$=\text{e}^{\sqrt{\text{x}}\log\text{a}}\frac{\log\text{a}}{2\sqrt{\text{x}}}$

$=\frac{\text{a}^{\sqrt{\text{x}}}}{2\sqrt{\text{x}}}\log_\text{e}\text{a}$

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 Derivatives→Derivatives — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Sketch the graphs of the following functions:
$\text{f(x)}=\cot2\text{x}$
→
Prove that:
$\cos10^\circ\cos30^\circ\cos50^\circ\cos70^\circ=\frac{3}{16}$
→
Show that the plane ax+ by+ cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio $=\frac{\text{ax}_1+\text{by}_1+\text{cz}_1+d}{\text{ax}_2+\text{by}_2+\text{cz}_2+d}.$
→
Use the Principle of Mathematical Induction in the following Exercis.
A sequence b0, b1, b2, ...... is defined by letting b0 = 5 and bk = 4 + bk-1 for all natural numbers k. Show that bn = 5 + 4n for all natural number n using mathematical induction.
→
Show that the point $(\text{x},\ \text{y})$ given by $\text{x}=\frac{2\text{at}}{1+\text{t}^2}$ and $\text{y}=\text{a}\Big(\frac{1-\text{t}^2}{1+\text{t}^2}\Big)$2 lies on a circle for all real values of t such that $-1\leq\text{t}\leq1,$ where a is any given real number.
→
Show that:

$\sin(\text{B}-\text{C})\cos(\text{A}-\text{D})+\sin(\text{C}-\text{A})\\\cos(\text{B}-\text{D})+\sin(\text{A}-\text{B})\cos(\text{C}-\text{D})=0$

→
Solve the following equations:
$\sin\text{x}+\sin2\text{x}+\sin3\text{x}+\sin4\text{x}=0$
→
Differentiate the following from first principle:

$(\text{x}+2)^3$

→
A and B are two events such that P(A) = 0.54, P(B) = 0.69 and $P(A \cap B) = 0.35$. Find
  1. $P(A \cup B)$
  2. $P(A'\cap B')$
  3. $P(A\cap B')$
  4. $P( B\cap A')$
→
Find the complex number satisfying the equation $\text{z}+\sqrt{2}|(\text{z}+1)|+\text{i}=0$
→