"Find the derivative of the Function: P(x)=(x−3cos⁡x)(x+3cos⁡x)" was answered by our experts

Chaya Galloway

Answered question

2021-01-15

Find the derivative of the Function:

P (x) = (x - 3 \cos x) (x + 3 \cos x)

Answer & Explanation

dessinemoie

Skilled2021-01-16Added 90 answers

P (x) = (x - 3 \cos x) (x + 3 \cos x) = x^{2} - 9 \cos^{2} (x)

\frac{d p}{d x} = 2 x - 9 d (\cos^{2} x) = 2 x - 9 \cdot 2 \cos x \cdot d (\cos x) =

= 2 x - 9 \cdot 2 \cos x (- \sin x)

= 2 x + 9 \cdot 2 \sin x \cos x

= 2 x + 9 \sin (2 x)

you can use the formula

(a - b) (a + b) = a^{2} - b^{2}

2 \sin x \cos x = \sin (2 x)

or you can use the product rule

\frac{d p}{d x} = (d (x - 3 \cos x)) (x + 3 \cos x) + (x - 3 \cos x) (d (x + 3 \cos x)) =

(1 + 3 \sin x) (x + 3 \cos x) + (x - 3 \cos x) (1 - 3 \sin x) =

x + 3 \cos x + 3 x \sin x + 9 \sin x \cos x + x - 3 \cos x - 3 x \sin x + 9 \sin x \cos x =

2 x + 18 \sin x \cos x = 2 x + 9 \cdot 2 \sin x \cos x = 2 x + 9 \sin (2 x)

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