Differentiate the following w.r.t. x: (x+1)^2(x+2)^3(x+3)^4 - Vidyadip

Question

Differentiate the following w.r.t. x:
$(\text{x}+1)^2(\text{x}+2)^3(\text{x}+3)^4$

✓

Answer

Let $\text{y}=(\text{x}+1)^2(\text{x}+2)^3(\text{x}+3)^4$
$\therefore\ \log\text{y}=\log\big\{(\text{x}+1)^2\cdot(\text{x}+2)^3(\text{x}+3)^4\big\}$
$=\log(\text{x}+1)^2+\log(\text{x}+2)^3+\log(\text{x}+3)^4$
and $\frac{\text{d}}{\text{dy}}\log\text{y}\cdot\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big[2\log(\text{x}+1)\big]\\+\frac{\text{d}}{\text{dx}}\big[3\log(\text{x}+2)\big]+\frac{\text{d}}{\text{dx}}\big[4\log(\text{x}+3)\big]$
$\frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\frac{2}{(\text{x}+1)}\cdot\frac{\text{d}}{\text{dx}}(\text{x}+1)+3\cdot\frac{1}{(\text{x}+2)}\cdot\\\frac{\text{d}}{\text{dx}}(\text{x}+2)+4\cdot\frac{1}{(\text{x}+3)}\cdot\frac{\text{d}}{\text{dx}}(\text{x}+3)$$\Big[\because\frac{\text{d}}{\text{dx}}(\log\text{x})=\frac{1}{\text{x}}\Big]$
$=\Big[\frac{2}{\text{x}+1}+\frac{3}{\text{x}+2}+\frac{4}{\text{x}+3}\Big]$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{2}{\text{x}+1}+\frac{3}{\text{x}+2}+\frac{4}{\text{x}+3}\Big]$
$=(\text{x}+1)^2\cdot(\text{x}+2)^3\cdot(\text{x}+3)^4\Big[\frac{2}{\text{x}+1}+\frac{3}{\text{x}+2}+\frac{4}{\text{x}+3}\Big]$
$=(\text{x}+1)^2\cdot(\text{x}+2)^3\cdot(\text{x}+3)^4$
$\bigg[\frac{2(\text{x}+2)(\text{x}+3)+3(\text{x}+1)(\text{x}+3)+4(\text{x}+1)(\text{x}+2)}{(\text{x}+1)(\text{x}+2)(\text{x}+3)}\bigg]$
$=\frac{(\text{x}+1)^2(\text{x}+2)^3(\text{x}+3)^4}{(\text{x}+1)(\text{x}+2)(\text{x}+3)}$
$\big[2(\text{x}^2+5\text{x}+6)+3(\text{x}^2+4\text{x}+3)+4(\text{x}^2+3\text{x}+2)\big]$
$=(\text{x}+1)(\text{x}+2)^2(\text{x}+3)^3$
$\big[2\text{x}^2+10\text{x}+12+3\text{x}^2+12\text{x}+9+4\text{x}^2+12\text{x}+8\big]$
$=(\text{x}+1)(\text{x}+2)^2(\text{x}+3)^3\big[9\text{x}^2+34\text{x}+29\big]$

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 "4 Marks" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equation
$\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}=\tan^{-1}\text{x},\text{x}\neq0$

At what points on the curve y = x² - 4x + 5 is the tangent perpendicular to the line 2y + x = 7?

Using integration, find the area of the region enclosed by the parabola y = 3x² and the line 3x - y + 6 = 0.

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = the first throw results in head,
B = the last throw results in tail.

Find the equation of the plane which contains the line of intersection of the planes $\overrightarrow{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})-4=0,\overrightarrow{\text{r}}\cdot(\hat{\text{2i}}+\hat{\text{j}}-\hat{\text{k}})+5=0$and which is perpendicular to the plane$\overrightarrow{\text{r}}\cdot(5\hat{\text{i}}+3\hat{\text{j}}-6\hat{\text{k}})+8=0.$

Evaluate the following integrals:
$\int\frac{1}{\sin^4\text{x}\cos^2\text{x}}\text{dx}$

If $\lim\limits_{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}$ exists finitely, write the value of $\lim\limits_{\text{x}\rightarrow{\text{c}}}\text{f(x)}.$

If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$.

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}})+9=0.$

A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him ₹ 360 and a manually operated sewing machine ₹ 240. He can sell the sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as an LPP and solve it graphically.