derivative of 5x^2cos(7x) — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} (5 x^{2} \cdot \cos (7 x))

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

\frac{d}{d x} (5 x^{2} \cdot \cos (7 x))

5 \cdot \frac{d}{d x} (x^{2} \cdot \cos (7 x))

5 \cdot \frac{d}{d x} (x^{2} \cdot \cos (7 x))

We differentiate

u1

x^{2}

and

v2

\cos (7 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 perform the differentiation

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 differentiate

v2

\frac{d}{d x} (\cos (7 x))

We differentiate using:

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

(where f =

7 x

)

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

\frac{d}{d x} (\cos (7 x))

- \frac{d}{d x} (7 x) \cdot \sin (7 x)

- \frac{d}{d x} (7 x) \cdot \sin (7 x)

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

\frac{d}{d x} (7 x)

7

- 7 \cdot \sin (7 x)

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

2 x \cdot \cos (7 x) + x^{2} \cdot (- 7 \cdot \sin (7 x))

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

x^{2} \cdot (-7) \cdot \sin (7 x)

- 7 x^{2} \cdot \sin (7 x)

2 x \cdot \cos (7 x) - 7 x^{2} \cdot \sin (7 x)

5) Then you can resume the differentiation:

ⅰ

\frac{d}{d x} (x^{2} \cdot \cos (7 x))

2 x \cdot \cos (7 x) - 7 x^{2} \cdot \sin (7 x)

5 \cdot (2 x \cdot \cos (7 x) - 7 x^{2} \cdot \sin (7 x))

ⓘ Expand by multiplying each term inside the parentheses:

5 \cdot (2 x \cdot \cos (7 x) - 7 x^{2} \cdot \sin (7 x))

10 x \cdot \cos (7 x) - 35 x^{2} \cdot \sin (7 x)

10 x \cdot \cos (7 x) - 35 x^{2} \cdot \sin (7 x)

This expression can be factorized into:

x \cdot (10 \cdot \cos (7 x) - 35 x \cdot \sin (7 x))

Let's factor in

ℝ

(10 \cdot \cos (7 x) - 35 x \cdot \sin (7 x))

We factor out

5

, the greatest common divisor of the coefficients:

10 \cdot \cos (7 x) - 35 x \cdot \sin (7 x)

5 \cdot (2 \cdot \cos (7 x) - 7 x \cdot \sin (7 x))

Factor identified:

2 \cdot \cos (7 x) - 7 x \cdot \sin (7 x)

▶Derivative:

5 x \cdot (2 \cdot \cos (7 x) - 7 x \cdot \sin (7 x))