derivative of sin(3x^2cos(2x)) — step-by-step solution

1) Let's perform the differentiation:

\frac{d}{d x} (\sin (3 x^{2} \cdot \cos (2 x)))

We differentiate using:

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

(where f =

3 x^{2} \cdot \cos (2 x)

)

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (3 x^{2} \cdot \cos (2 x)))

\frac{d}{d x} (3 x^{2} \cdot \cos (2 x)) \cdot \cos (3 x^{2} \cdot \cos (2 x))

\frac{d}{d x} (3 x^{2} \cdot \cos (2 x)) \cdot \cos (3 x^{2} \cdot \cos (2 x))

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

\frac{d}{d x} (3 x^{2} \cdot \cos (2 x))

3 \cdot \frac{d}{d x} (x^{2} \cdot \cos (2 x))

3 \cdot \frac{d}{d x} (x^{2} \cdot \cos (2 x)) \cdot \cos (3 x^{2} \cdot \cos (2 x))

We differentiate

u1

x^{2}

and

v2

\cos (2 x)

using:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's differentiate

u1

\frac{d}{d x} (x^{2})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

2 x

3) Let's perform the differentiation

v2

\frac{d}{d x} (\cos (2 x))

We differentiate using:

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

(where f =

2 x

)

ⓘ The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (2 x))

- \frac{d}{d x} (2 x) \cdot \sin (2 x)

- \frac{d}{d x} (2 x) \cdot \sin (2 x)

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

\frac{d}{d x} (2 x)

2

- 2 \cdot \sin (2 x)

4) We evaluate the resulting derivative (

u1 \cdot v2

)':

2 x \cdot \cos (2 x) + x^{2} \cdot (- 2 \cdot \sin (2 x))

ⓘ The minus sign is placed in front of the product:

x^{2} \cdot (-2) \cdot \sin (2 x)

- 2 x^{2} \cdot \sin (2 x)

2 x \cdot \cos (2 x) - 2 x^{2} \cdot \sin (2 x)

5) Then you can resume the derivation:

ⅰ

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

2 x \cdot \cos (2 x) - 2 x^{2} \cdot \sin (2 x)

3 \cdot (2 x \cdot \cos (2 x) - 2 x^{2} \cdot \sin (2 x)) \cdot \cos (3 x^{2} \cdot \cos (2 x))

ⓘ Expand by multiplying each term inside the parentheses:

3 \cdot (2 x \cdot \cos (2 x) - 2 x^{2} \cdot \sin (2 x))

6 x \cdot \cos (2 x) - 6 x^{2} \cdot \sin (2 x)

(6 x \cdot \cos (2 x) - 6 x^{2} \cdot \sin (2 x)) \cdot \cos (3 x^{2} \cdot \cos (2 x))

ⓘ Expand by multiplying each term inside the parentheses:

(6 x \cdot \cos (2 x) - 6 x^{2} \cdot \sin (2 x)) \cdot \cos (3 x^{2} \cdot \cos (2 x))

6 x \cdot \cos (2 x) \cdot \cos (3 x^{2} \cdot \cos (2 x)) - 6 x^{2} \cdot \sin (2 x) \cdot \cos (3 x^{2} \cdot \cos (2 x))

6 x \cdot \cos (2 x) \cdot \cos (3 x^{2} \cdot \cos (2 x)) - 6 x^{2} \cdot \sin (2 x) \cdot \cos (3 x^{2} \cdot \cos (2 x))

6 x \cdot \cos (2 x) \cdot \cos (3 x^{2} \cdot \cos (2 x)) - 6 x^{2} \cdot \cos (3 x^{2} \cdot \cos (2 x)) \cdot \sin (2 x)

You can write the expression in factored form:

x \cdot \cos (3 x^{2} \cdot \cos (2 x)) \cdot (6 \cdot \cos (2 x) - 6 x \cdot \sin (2 x))

Let's factorize in

ℝ

(6 \cdot \cos (2 x) - 6 x \cdot \sin (2 x))

We factor out

6

, the greatest common divisor of the coefficients:

6 \cdot \cos (2 x) - 6 x \cdot \sin (2 x)

6 \cdot (\cos (2 x) - x \cdot \sin (2 x))

Factor found:

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

▶Derivative:

6 x \cdot \cos (3 x^{2} \cdot \cos (2 x)) \cdot (\cos (2 x) - x \cdot \sin (2 x))