Find the second-order derivative of the funct | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $y = x^{3} \log x$. First,we find the first-order derivative using the product rule: $\frac{dy}{dx} = \frac{d}{dx}(x^{3}) \cdot \log x + x^{3}

Vedclass · Accepted Answer

$x(5 + 6 \log x)$

Vedclass · Answer

(A) Let $y = x^{3} \log x$. First,we find the first-order derivative using the product rule: $\frac{dy}{dx} = \frac{d}{dx}(x^{3}) \cdot \log x + x^{3} \cdot \frac{d}{dx}(\log x)$ $= 3x^{2} \log x + x^{3} \cdot \frac{1}{x}$ $= 3x^{2} \log x + x^{2} = x^{2}(1 + 3 \log x)$. Now,we find the second-order derivative: $\frac{d^{2}y}{dx^{2}} = \frac{d}{dx}[x^{2}(1 + 3 \log x)]$ $= \frac{d}{dx}(x^{2}) \cdot (1 + 3 \log x) + x^{2} \cdot \frac{d}{dx}(1 + 3 \log x)$ $= 2x(1 + 3 \log x) + x^{2} \cdot \frac{3}{x}$ $= 2x + 6x \log x + 3x$ $= 5x + 6x \log x$ $= x(5 + 6 \log x)$.

Find the second-order derivative of the function $f(x) = x^{3} \log x$.

Similar Questions

If $f(x)=x^4-x^3+7x^2+14$,then what is the value of $f^{\prime \prime}(5)$?

If $y=3 e^{2 x}+2 e^{3 x},$ prove that $\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$.

If $y = a^x \cdot b^{2x-1}$,then $\frac{d^2 y}{d x^2}$ is equal to

If $y = \frac{ax + b}{cx + d}$, then $2 \frac{dy}{dx} \frac{d^3y}{dx^3}$ is equal to

If $f(x) = 3x^3 + 2x^2 f'(1) + x f''(2) + f'''(3)$,then $f(x) = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module