Derivative of `x^2*sin(x)`

Step-by-step differentiation

Answer

\frac{d}{d x} [x^{2} sin (x)]

2 x \cdot sin (x) + x^{2} cos (x)

Step-by-step solution

1Product Rule
$\frac{d}{d x} [x^{2} sin (x)] = \frac{d}{d x} [x^{2}] \cdot sin (x) + x^{2} \cdot \frac{d}{d x} [sin (x)]$
Let $f = x^{2}$ and $g = sin (x)$ . The product rule says $(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}$ — find each factor's derivative, then assemble.
↳Find f′: Differentiate the first factor, $f = x^{2}$ :
2Power Rule
$\frac{d}{d x} [x^{2}] = 2 \cdot x^{2 - 1}$
The exponent is $2$ . Bring it down in front as a coefficient, then reduce the power by $1$ .
↳Find g′: Differentiate the second factor, $g = sin (x)$ :
3Derivative of sin
$\frac{d}{d x} [sin (x)] = cos (x)$
The derivative of $sin (x)$ is $cos (x)$ — a standard rule.
=Combine
$\frac{d}{d x} [x^{2} sin (x)] = 2 x \cdot sin (x) + x^{2} \cdot cos (x) = 2 x \cdot sin (x) + x^{2} cos (x)$
With $f^{'} = 2 x$ and $g^{'} = cos (x)$ , substitute into $f^{'} \cdot g + f \cdot g^{'}$ and simplify.
✓Final Answer
$\frac{d}{d x} [x^{2} sin (x)] = 2 x \cdot sin (x) + x^{2} cos (x)$
The first derivative simplifies to $2 x \cdot sin (x) + x^{2} cos (x)$ .

Understanding this derivative

Differentiating x²·sin(x) exercises 3 distinct techniques — the product rule, the power rule and the trigonometric derivatives. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

Because two expressions are multiplied together, the product rule applies: differentiate each factor in turn while keeping the other fixed, then add the results. Note that the derivative of a product is not simply the product of the derivatives. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. The trigonometric derivatives follow the standard cycle: sine differentiates to cosine, cosine to negative sine, and tangent to secant squared. The sign on the cosine derivative is a frequent source of errors, so it is worth double-checking.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of x^2*sin(x)

Step-by-step solution

Understanding this derivative

Derivative of `x^2*sin(x)`