We have an answer to "What is the derivative of xsin⁡x"

High School
Geometry
Trigonometry
Trigonometric Functions

rheisf

Answered question

2021-12-29

What is the derivative of

x \sin x

Answer & Explanation

Bernard Lacey

Beginner2021-12-30Added 30 answers

\frac{d y}{d x} = x \cos x + \sin x

Explanation:
We have:

y = x \sin x

Which is the product of two functions, and so we apply the Product Rule for Differentiation:

\frac{d}{d x} (u v) = u \frac{d v}{d x} + \frac{d u}{d x} v, or {(u v)}^{'} = (d u) v + u (d v)

I was taught to remember the rule in words; "The first times the derivative of the second plus the derivative of the first times the second ".
So with

y = x \sin x

{\begin{cases} Let & u = x & \Rightarrow \frac{d u}{d x} = 1 \\ And & v = \sin x & \Rightarrow \frac{d v}{d x} = \cos x \end{cases}

Then:

\frac{d}{d x} (u v) = u \frac{d v}{d x} + \frac{d u}{d x} v

Gives us:

\frac{d}{d x} (x \sin x) = (x) (\cos x) + (1) (\sin x)

∴ \frac{d y}{d x} = x \cos x + \sin x

If you are new to Calculus then explicitly substituting u and v can be quite helpful, but with practice these steps can be omitted, and the product rule can be applied as we write out the solution.

Maricela Alarcon

Beginner2021-12-31Added 28 answers

This is a function which is in the form,

y = f (x) g (x)

Its

Vasquez

Expert2022-01-08Added 669 answers

Finally found the answer to this difficult question, I'm happy

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