Question
Tentukan turunan pertama dari fungsi \[ y=3 x^{2} \cos 6 x \] A. \( 6 x \cos 6 x-18 x^{2} \sin 6 x \) B. \( 6 x \cos 6 x+18 x^{2} \sin 6 x \) C. \( 6 x \cos 6 x-12 x^{2} \sin 6 x \) D. \( 6 x \cos 6 x+12 x^{2} \sin 6 x \) E. \( 3 x \cos 6 x-12 x^{2} \sin 6 x \)
Ask by Mcfarlane Wilson. in Indonesia
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( y = 3x^2 \cos 6x \) is \( 6x \cos 6x - 18x^2 \sin 6x \). So, the correct choice is A.
Solution
To find the derivative of
y = 3x² cos 6x,
we use the product rule. Let
u(x) = 3x² and v(x) = cos 6x.
First, compute the derivatives:
u'(x) = 6x,
v'(x) = –sin 6x · 6 = –6 sin 6x (using the chain rule).
Now, applying the product rule:
y' = u'(x)·v(x) + u(x)·v'(x)
= 6x cos 6x + 3x² (–6 sin 6x)
= 6x cos 6x – 18x² sin 6x.
Thus, the correct answer is option A.
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Simplify this solution Bonus Knowledge
Untuk menghitung turunan pertama dari fungsi \( y = 3 x^2 \cos(6x) \), kita perlu menggunakan aturan produk, di mana \( u = 3x^2 \) dan \( v = \cos(6x) \). Langkah pertama adalah mencari turunan dari \( u \) dan \( v \): - \( u' = 6x \) - \( v' = -\sin(6x) \cdot 6 = -6\sin(6x) \) Kemudian, dengan aturan produk \( (uv)' = u'v + uv' \): \[ y' = (6x)(\cos(6x)) + (3x^2)(-6\sin(6x)) \] \[ y' = 6x \cos(6x) - 18x^2 \sin(6x) \] Jadi, turunan pertama dari fungsi tersebut adalah: **A. \( 6 x \cos 6 x - 18 x^{2} \sin 6 x \)**.
