How to differentiate y = x 2 cos x ?

Question

How to differentiate       y    =          x              2                  cos          x                    ?

Hayden Dudley · Accepted Answer

The derivative for this is found by the chain rule.
Let,

y = f (x) = x^{2 \cos x}

.

∴ \frac{d y}{d x} = \frac{d}{d x} (x^{2 \cos x}) \cdot \frac{d}{d x} (2 \cos x)

.

∴ \frac{d y}{d x} = (2 \cos x \cdot x^{2 \cos x - 1}) \cdot (- 2 \sin x)

.
Hence,

∴ \frac{d y}{d x} = - 4 \sin x \cos x \cdot x^{2 \cos x - 1}

.

criticoneitor5nq · Accepted Answer

Use some version of logarithmic differentiation.
Rewriting logarithmic difference as e to a power is my current preferred form.

y = x^{2 \cos x} = e^{2 \cos x \ln x}

Thus, use

\frac{d}{d x} (e^{u}) = e^{u} \frac{d u}{d x}

to get

\frac{d y}{d x} = e^{2 \cos x \ln x} \cdot \frac{d}{d x} (2 \cos x \ln x)

= x^{2 \cos x} \cdot [\frac{2 \cos x}{x} - 2 \sin x \ln x]

How do you differentiate y=x^(2 cos x)?