186k views
3 votes
Differentiate f(x)=x^2sin(x)

User Scabbage
by
5.2k points

1 Answer

3 votes

Answer:


\displaystyle \large{f\prime(x) = 2x \sin (x) + x^2 \cos (x)}

Explanation:

We are given a function:


\displaystyle \large{f(x) = x^2 \sin (x)}

Notice that there are two functions multiplying each other. Recall the product rules.


\displaystyle \large{y=h(x)g(x) \to y\prime = h\prime (x) g(x) + h(x)g\prime (x)}

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:


\displaystyle \large{f(x) = x^2 \sin (x) \to f\prime (x) = (x^2)\prime \sin(x) + x^2 (\sin (x))\prime}

Recall the power rules and differentiation of sine.


\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) }

Therefore, from differentiating function,


\displaystyle \large{f\prime(x) = 2x \sin (x) + x^2 \cos (x)}

And we are done!

User Irfan Gul
by
5.0k points