Best solutions to - "Find the derivatives of the function defined as..."

Lorraine Harvey

Answered question

2021-12-11

Find the derivatives of the function defined as follows.

y = - 6 x^{4} \sin 6 x

Buck Henry

Beginner2021-12-12Added 33 answers

Step 1
Given equation −

y = - 6 x^{4} \sin (6 x)

...(1)
Differentiating on both sides with respect to x −

\frac{d y}{d x} = - 6 \frac{d}{d x} [x^{4} \sin (6 x)]

\frac{d y}{d x} = - 6 [x^{4} \frac{d}{d x} \sin (6 x) + \sin (6 x) \frac{d}{d x} x^{4}]

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

\frac{d y}{d x} = - 6 [x^{4} \cos (6 x) \frac{d}{d x} (6 x) + \sin (6 x) (4) x^{(4 - 1)}]

[\frac{d}{d x} x^{n} = n x^{(n - 1)}]

\frac{d y}{d x} = - 6 [x^{4} \cos (6 x) (6) + 4 x^{3} \sin (6 x)]

\frac{d y}{d x} = - 6 [6 x^{4} \cos (6 x) + 4 x^{3} \sin (6 x)]

Step 2
Hence, the derivative of the given function is −

\frac{d y}{d x} = - 6 [6 x^{4} \cos (6 x) + 4 x^{3} \sin (6 x)]

\frac{d y}{d x} = - 36 x^{4} \cos (6 x) - 24 x^{3} \sin (6 x)

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