LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question
Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES4 Marks
Question
Find the derivative of the following functions: $ 2 \tan\text{x} – 7\sec\text{x}$
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Answer
Let $\text{f}(\text{x})= 2 \tan\text{x} – 7\sec\text{x}$ Accordingly, from the first principle, $\text{f}'(\text{x})=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\big[2\tan(\text{x}+\text{h})-7\sec(\text{x}+\text{h})-2\tan\text{x}+7\sec\text{x}\big]$$=\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\big[2\big\{\tan(\text{x}+\text{h})-\tan\text{x}\big\}-7\big\{\sec(\text{x}+\text{h})-\sec\text{x}\big\}\big]$
$=2\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\big[\big\{\tan(\text{x}+\text{h})-\tan\text{x}\big\}-7\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\big\{\sec(\text{x}+\text{h})-\sec\text{x}\big\}\big]$
$=2\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\bigg[\frac{\sin(\text{x}+\text{h})}{\cos(\text{x}+\text{h})}-\frac{\sin\text{x}}{\cos\text{x}}\bigg]-7\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\Bigg[\frac{1}{\cos(\text{x}+\text{h})}-\frac{1}{\cos\text{x}}\Bigg]$
$$$=2\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\bigg[\frac{\sin(\text{x}+\text{h})\cos\text{x}-\sin\text{x}\cos(\text{x}+\text{h})}{\cos\text{x}\cos(\text{x}+\text{h})}\bigg]-7\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\Bigg[\frac{{\cos\text{x}}-{\cos(\text{x}+\text{h})}}{{\cos\text{x}}\cos(\text{x}+\text{h})}\Bigg]$
$=2\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\bigg[\frac{\sin(\text{x}+\text{h}-\text{x})}{\cos\text{x}\cos(\text{x}+\text{h})}\bigg]-7\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\Bigg[\frac{-2\sin\bigg(\frac{\text{x}+\text{x}+\text{h}}{2}\bigg)+\sin\bigg(\frac{\text{x}+\text{x}-\text{h}}{2}\bigg)}{{\cos\text{x}}\cos(\text{x}+\text{h})}\Bigg]$ $=2\lim\limits_{\text{h}\rightarrow0}\bigg[\bigg(\frac{\sin\text{x}}{\text{h}}\bigg)\frac{1}{\cos\text{x}\cos(\text{x}+\text{h})}\bigg]-7\lim\limits_{\text{h}\rightarrow0}\frac{1}{\text{h}}\Bigg[\frac{-2\sin\bigg(\frac{2\text{x}+\text{h}}{2}\bigg)+\sin\bigg(-\frac{\text{h}}{2}\bigg)}{{\cos\text{x}}\cos(\text{x}+\text{h})}\Bigg]$$=2\bigg(\lim\limits_{\text{h}\rightarrow0}\frac{\sin\text{x}}{\text{h}}\bigg)\bigg(\lim\limits_{\text{h}\rightarrow0}\frac{1}{\cos\text{x}\cos(\text{x}+\text{h})}\bigg)-7\Bigg(\lim\limits_{\frac{\text{h}}{2}\rightarrow0}\frac{\sin\frac{\text{h}}{\text{h}}{2}}{2}\Bigg)\Bigg(\lim\limits_{\text{h}\rightarrow0}\frac{\sin\bigg(\frac{2\text{x}+\text{h}}{2}\bigg)}{{\cos\text{x}}\cos(\text{x}+\text{h})}\Bigg)$
$=2\bigg(\lim\limits_{\text{h}\rightarrow0}\frac{\sin\text{x}}{\text{h}}\bigg)\bigg(\lim\limits_{\text{h}\rightarrow0}\frac{1}{\cos\text{x}\cos(\text{x}+\text{h})}\bigg)-7\Bigg(\lim\limits_{\frac{\text{h}}{2}\rightarrow0}\frac{\sin\frac{\text{h}}{2}}{\frac{\text{h}}{2}}\Bigg)\Bigg(\lim\limits_{\text{h}\rightarrow0}\frac{\sin\bigg(\frac{2\text{x}+\text{h}}{2}\bigg)}{{\cos\text{x}}\cos(\text{x}+\text{h})}\Bigg)$$=2.1.\frac{1}{\cos\text{x}\cos\text{x}}-7.1\bigg(\frac{\sin\text{x}}{\cos\text{x}\cos\text{x}}\bigg)$
$=2\sec^2\text{x}-7\sec\text{x}\tan\text{x}$
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