Download App

Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsDerivatives5 Marks
Question
Differentiate the following from first principle:$\text{x}^3+4\text{x}^2+3\text{x}+2$
✓

Answer

We have,$\text{f(x)}=\text{x}^3+4\text{x}^2+3\text{x}+2$
$\because\text{f}'\text{(x)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{x+h})-\text{f}(\text{x})}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{(\text{x+h})^3+4(\text{x+h})^2+3(\text{x+h})+2-(\text{x}^3+4\text{x}^2+3\text{x}+2)}{\text{h}}$
On solving we get,$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{x}^3+\text{h}^3+3\text{x}^2\text{h}+3\text{h}^2\text{x}+4\text{x}^2+4\text{h}^2+8\text{hx}+3\text{x}+3\text{h}+2-\text{x}^3-4\text{x}^2-3\text{x}-2}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{3\text{x}^2\text{h}+3\text{xh}^2+\text{h}^3+4\text{h}^2+8\text{hx}+3\text{h}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}3\text{x}^2+3\text{xh}+\text{h}^2+4\text{h}+8\text{x}+3$
$=3\text{x}^2+8\text{x}+3$

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→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

$\text{a}^2=(\text{b + c})^2-4\text{bc}\cos^2\frac{\text{A}}{2}$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(3+\text{x})-\sin(3-\text{x})}{\text{x}}$
→
Prove that:
$\sin\alpha+\sin\beta+\sin\gamma-\sin(\alpha+\beta+\gamma)\\=4\sin\Big(\frac{\alpha+\beta}{2}\Big)\sin\Big(\frac{\beta+\gamma}{2}\Big)\sin\Big(\frac{\gamma+\alpha}{2}\Big)$
→
If angle $\theta$ is divided into two parts such that the tangents of one part is $\lambda$ times the tangent of other,and $\phi$ is their difference,the show that $\sin\theta=\frac{\lambda+1}{\lambda-1}\sin\phi$
→
Match the following:
$a.$ If $E_1$ and $E_2$ are the two mutually exclusive events $i.$ $\text{E}_1\cap\text{E}_2=\text{E}_1$
$b.$ If $E_1$ and $E_2$ are the mutually exclusive and exhaustive events $ii.$ $(\text{E}_1-\text{E}_2)\cup(\text{E}_1\cap\text{E}_2)=\text{E}_1$
$c.$ If $E_1$ and $E_2$ have common outcomes, then $iii.$ $\text{E}_1\cap\text{E}_2=\phi,\text{ E}_1\cup\text{E}_2=\text{S}$
$d.$ If $E_1$ and $E_2$ are two events such that $\text{E}_1\subset\text{E}_2$ $iv.$ $\text{E}_1\cap\text{E}_2=\phi$
→
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.
→
Find the term independent of x in the expansion of the following expressions:
$\Big(\frac{3\text{x}^{2}}{2}-\frac{1}{3\text{x}}\Big)^{9}$
→
Show that the tangent of an angle between the lines $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1$ and $\frac{\text{x}}{\text{a}}-\frac{\text{y}}{\text{b}}=1$ is $\frac{2\text{ab}}{\text{a}^2-\text{b}^2}.$
→
Use the Principle of Mathematical Induction in the following Exercis.
Prove that for all $\text{n}\in\text{N},\cos\alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+\ ....\ +\cos\big[\alpha+(\text{n}-1)\beta\big]$$=\frac{\cos\Big[\alpha+\big(\frac{\text{n}-1}{2}\big)\beta\Big]\sin\big(\frac{\text{n}\beta}{2}\big)}{\sin\frac{\beta}{2}}$
→
Differentiate the following from first principle$3^{\text{x}^2}$
→