"Differentiate.f(x)=x2sin⁡x" solved and explained

cinearth3

Open question

2022-08-18

Differentiate.

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

Answer & Explanation

Bryanna Villarreal

Beginner2022-08-19Added 5 answers

Step 1
We are asked to differentiate the given function; to do this we have to use the power rule and remember our trid identities when it comes to taking the derivative.

f (x) = x^{2} \times \sin (x)

Step 2
We will have to remember that the derivative of

\sin (x)

\cos (x)

f^{'} (x) = x^{2} \times \sin (x)

= x^{2} \times \frac{d}{d x} (\sin (x)) + \frac{d}{d x} (x^{2}) \times \sin (x)

= x^{2} \times \cos (x) + 2 \times x \times \sin (x)

Braeden Valenzuela

Beginner2022-08-20Added 3 answers

Step 1
Let

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

then

f (x) = g (x) \times h (x)

The derivative of this function is given by

f^{'} (x) = (g^{'} (x) \times h (x)) + (h^{'} (x) \times g (x))

The derivative of

g (x)

x^{2}

g^{'} (x) = 2 \times x^{2 - 1} = 2 x

The derivative of

h (x)

\sin x

h^{'} (x) = \cos x

Applying the product rule:

f^{'} (x) = (g^{'} (x) \times h (x)) + (h^{'} (x) \times g (x))

f^{'} (x) = (2 x (\sin x)) + (x^{2} (\cos x))

f^{'} (x) = 2 x \sin x + x^{2} \cos x

Hence, the derivative of

y = (x^{2}) (\sin x)

y^{'} = 2 x \sin x + x^{2} \cos x

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