Find the derivative of the function at the given value of x using a graphing utility. If necessary, round to four decimal
places.
-$$f ( x ) = x \sin ( 10 x ) ; x = 8$$
A) -9.8247
B) -7,950.7128
C) -8.8297
D) 7,950.7128

Question

Quizplus · Accepted Answer

The derivative of $f(x) = x\sin(10x)$ can be found using the product rule of differentiation, which states if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$. Applying this to the given function, where $u(x) = x$ and $v(x) = \sin(10x)$, we get $f'(x) = 1 \cdot \sin(10x) + x \cdot 10\cos(10x)$. Evaluating this derivative at $x = 8$ and using a graphing utility or a calculator to approximate the value, we get approximately -9.8247.

Find the Derivative of the Function at the Given Value f(x)=xsin⁡(10x);x=8f ( x ) = x \sin ( 10 x ) ; x = 8f(x)=xsin(10x);x=8

Find the Derivative of the Function at the Given Value $f ( x ) = x \sin ( 10 x ) ; x = 8$