derivative of sin(3x+cos(3x)) — step-by-step solution

Let's differentiate:

\frac{d}{d x} (\sin (3 x + \cos (3 x)))

We differentiate using:

\frac{d}{d x} (\sin (f)) = \frac{d}{d x} (f) \cdot \cos (f)

(where f =

3 x + \cos (3 x)

)

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (3 x + \cos (3 x)))

\frac{d}{d x} (3 x + \cos (3 x)) \cdot \cos (3 x + \cos (3 x))

\frac{d}{d x} (3 x + \cos (3 x)) \cdot \cos (3 x + \cos (3 x))

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (3 x + \cos (3 x))

\frac{d}{d x} (3 x) + \frac{d}{d x} (\cos (3 x))

(\frac{d}{d x} (3 x) + \frac{d}{d x} (\cos (3 x))) \cdot \cos (3 x + \cos (3 x))

We differentiate using:

\frac{d}{d x} (\cos (f)) = - \frac{d}{d x} (f) \cdot \sin (f)

(where f =

3 x

)

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (3 x)

3

■The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (3 x))

- \frac{d}{d x} (3 x) \cdot \sin (3 x)

(3 - \frac{d}{d x} (3 x) \cdot \sin (3 x)) \cdot \cos (3 x + \cos (3 x))

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (3 x)

3

(3 - 3 \cdot \sin (3 x)) \cdot \cos (3 x + \cos (3 x))

\cos (3 x + \cos (3 x)) \cdot (3 - 3 \cdot \sin (3 x))

Let's factor in

ℝ

(3 - 3 \cdot \sin (3 x))

We factor out

3

, the greatest common divisor of the coefficients:

3 - 3 \cdot \sin (3 x)

3 \cdot (1 - \sin (3 x))

Factor found:

1 - \sin (3 x)

▶Derivative:

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