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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives1 Mark
Question
Find the derivative of function sin (x + 1) from first principle.
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Answer

Here f (x) = sin (x + 1)
Then f (x + h) = sin (x + h + 1)
We know that $f{\text{'}}(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}$
$\Rightarrow \;f{\text{'}}(x) = \mathop {\lim }\limits_{h \to 0} \frac{{\sin (x + h + 1) - \sin (x + 1)}}{h}$
$= \mathop {\lim }\limits_{h \to 0} \frac{{2\cos \left( {\frac{{2x + h + 2}}{2}} \right)\sin \left( {\frac{h}{2}} \right)}}{h}$
$= \mathop {\lim }\limits_{h \to 0} \frac{{\cos \left( {x + 1 + \frac{h}{2}} \right)\sin \left( {\frac{h}{2}} \right)}}{{\left( {\frac{h}{2}} \right)}}$ = cos (x + 1)

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