If y=x^3 , Prove that d^4y dx^4 =6 x - Vidyadip

Question

If $\text{y}=\text{x}^3\log\text{x},$ Prove that $\frac{\text{d}^4\text{y}}{\text{dx}^4}=\frac{6}{\text{x}}$

✓

Answer

here,
$\text{y}=\text{x}^3\log\text{x},$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=3\text{x}^2\log{x}+\text{x}^3\times\frac{1}{\text{x}}$
$=3\text{x}^2\log{\text{x}}+\text{x}^2$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=6\text{x}\log\text{x}+3\text{x}^2\times\frac{1}{\text{x}}+2\text{x}$
$=6\text{x}\log\text{x}+5\text{x}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=6\log\text{x}+6\text{x}\times\frac{1}{\text{x}}+5=6\log\text{x}+11$
Differentiating w.r.t.x, we get

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 "Solve the Following Question.(5 Marks)" from Higher Order Derivatives→Higher Order Derivatives — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The function $y=a \log x+b x^2+x$ has extreme values at $x=1$ and $x=2$. Find $a$ and $b$.

$\int\frac{\text{x}+1}{\sqrt{2\text{x}+3}}\text{dx}$

Evaluate the following intregals:
$\int\frac{2\text{x}^2+7\text{x}-3}{\text{x}^2(2\text{x}+1)}\text{ dx}$

Compare the area under the curve $\text{y}=\cos^2\text{x}\text{ and }\text{y}=\sin^2\text{x}$ between $x = 0$ and $\text{x}=\pi.$

Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$

Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}})$ and the plane $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5.$

If $\text{y}=\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}},$ prove that $2\text{x}\frac{\text{dy}}{\text{dx}}=\sqrt{\text{x}}-\frac{1}{\sqrt{\text{x}}}$

Evaluate the following intregals:
$\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs. 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs. 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.