Answer:
[tex]\displaystyle \large{f\prime(x) = 2x \sin (x) + x^2 \cos (x)}[/tex]
Step-by-step explanation:
We are given a function:
[tex]\displaystyle \large{f(x) = x^2 \sin (x)}[/tex]
Notice that there are two functions multiplying each other. Recall the product rules.
[tex]\displaystyle \large{y=h(x)g(x) \to y\prime = h\prime (x) g(x) + h(x)g\prime (x)}[/tex]
You can let x^2 = h(x), sin(x) = g(x) or sin(x) = h(x), x^2 = g(x) as your desire but I’ll let h(x) = x^2 and g(x) = sin(x).
Therefore, from a function:
[tex]\displaystyle \large{f(x) = x^2 \sin (x) \to f\prime (x) = (x^2)\prime \sin(x) + x^2 (\sin (x))\prime}[/tex]
Recall the power rules and differentiation of sine.
[tex]\displaystyle \large{y = ax^n \to y\prime = nax^{n-1} \ \ \ \tt{for \ \ polynomial \ \ function}}\\ \displaystyle \large{y = \sin (x) \to y\prime = \cos (x) }[/tex]
Therefore, from differentiating function,
[tex]\displaystyle \large{f\prime(x) = 2x \sin (x) + x^2 \cos (x)}[/tex]
And we are done!
