derivative of (1-25x^2)^2arcsin(5x)

1) Let's perform the differentiation:

\frac{d}{d x} ({(1 - 25 x^{2})}^{2} \cdot \arcsin (5 x))

For

u1

{(1 - 25 x^{2})}^{2}

and

v2

\arcsin (5 x)

, use the differentiation rule:

\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} ({(1 - 25 x^{2})}^{2})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

1 - 25 x^{2}

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(1 - 25 x^{2})}^{2})

2 \cdot \frac{d}{d x} (1 - 25 x^{2}) \cdot (1 - 25 x^{2})

2 \cdot \frac{d}{d x} (1 - 25 x^{2}) \cdot (1 - 25 x^{2})

ⓘ Expand by multiplying each term inside the parentheses:

2 \cdot \frac{d}{d x} (1 - 25 x^{2}) \cdot (1 - 25 x^{2})

2 \cdot \frac{d}{d x} (1 - 25 x^{2}) - 50 \cdot \frac{d}{d x} (1 - 25 x^{2}) \cdot x^{2}

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

\frac{d}{d x} (1 - 25 x^{2})

0 + \frac{d}{d x} (- 25 x^{2})

2 \cdot (0 + \frac{d}{d x} (- 25 x^{2})) - 50 \cdot (0 + \frac{d}{d x} (- 25 x^{2})) \cdot x^{2}

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- 25 x^{2})

\frac{d}{d x} (- 25 x^{2})

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

\frac{d}{d x} (- 25 x^{2})

-25 \cdot \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 \cdot (-25 \cdot 2 x) - 50 \cdot (-25 \cdot 2 x) \cdot x^{2}

ⓘ Multiplications are done before additions and subtractions:

25 \cdot 2

(20 + 5) \cdot 2

20 \cdot 2 + 5 \cdot 2

40 + 10

50

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-50 \cdot (-25)

50 \cdot 25

50 \cdot (20 + 5)

50 \cdot 20 + 50 \cdot 5

1000 + 250

1250

■Multiplying powers with the same base adds the exponents:

x x^{2}

x^{3}

- 50 \cdot 2 x + 1250 \cdot 2 x^{3}

ⓘ Multiplications are done before additions and subtractions:

50 \cdot 2

100

1250 \cdot 2

(1000 + 200 + 50) \cdot 2

1000 \cdot 2 + 200 \cdot 2 + 50 \cdot 2

2000 + 400 + 100

2500

- 100 x + 2500 x^{3}

3) Let's differentiate

v2

\frac{d}{d x} (\arcsin (5 x))

Apply the differentiation rule:

\frac{d}{d x} (\arcsin (f)) = \frac{d}{d x} (f) \cdot \frac{1}{\sqrt{1 - f^{2}}}

with f =

5 x

ⓘ The derivative of arcsin(x) is 1/sqrt(1 - x^2):

\frac{d}{d x} (\arcsin (5 x))

\frac{\frac{d}{d x} (5 x)}{\sqrt{1 - {(5 x)}^{2}}}

\frac{\frac{d}{d x} (5 x)}{\sqrt{1 - {(5 x)}^{2}}}

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

\frac{d}{d x} (5 x)

5

■The power of a product is the product of the powers:

{(5 x)}^{2}

5^{2} \cdot x^{2}

\frac{5}{\sqrt{1 - 5^{2} \cdot x^{2}}}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

5^{2}

5 \cdot 5

25

\frac{5}{\sqrt{1 - 25 x^{2}}}

4) We evaluate the resulting derivative (

u1 \cdot v2

)':

(- 100 x + 2500 x^{3}) \cdot \arcsin (5 x) + {(1 - 25 x^{2})}^{2} \cdot \frac{5}{\sqrt{1 - 25 x^{2}}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

{(1 - 25 x^{2})}^{2} \cdot \frac{5}{\sqrt{1 - 25 x^{2}}}

\frac{{(1 - 25 x^{2})}^{2} \cdot 5}{\sqrt{1 - 25 x^{2}}}

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

\frac{{(1 - 25 x^{2})}^{2} \cdot 5}{\sqrt{1 - 25 x^{2}}}

5 \cdot {(1 - 25 x^{2})}^{2 - \frac{1}{2}}

(- 100 x + 2500 x^{3}) \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{2 - \frac{1}{2}}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

2 - \frac{1}{2}

\frac{2 \cdot 2 - 1}{2}

(- 100 x + 2500 x^{3}) \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{2 \cdot 2 - 1}{2}}

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 2

4

(- 100 x + 2500 x^{3}) \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{4 - 1}{2}}

ⓘ The numerator and the denominator are computed before the division:

4 - 1

3

(- 100 x + 2500 x^{3}) \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{3}{2}}

ⓘ Expand by multiplying each term inside the parentheses:

(- 100 x + 2500 x^{3}) \cdot \arcsin (5 x)

- 100 x \cdot \arcsin (5 x) + 2500 x^{3} \cdot \arcsin (5 x)

- 100 x \cdot \arcsin (5 x) + 2500 x^{3} \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{3}{2}}

5) Then you can resume the derivation:

ⅰ

\frac{d}{d x} ({(1 - 25 x^{2})}^{2} \cdot \arcsin (5 x))

- 100 x \cdot \arcsin (5 x) + 2500 x^{3} \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{3}{2}}

- 100 x \cdot \arcsin (5 x) + 2500 x^{3} \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{3}{2}}

2500 x^{3} \cdot \arcsin (5 x) - 100 x \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{3}{2}}

We factor out

5

, the greatest common divisor of the coefficients:

2500 x^{3} \cdot \arcsin (5 x) - 100 x \cdot \arcsin (5 x) + 5 \cdot {(1 - 25 x^{2})}^{\frac{3}{2}}

5 \cdot (500 x^{3} \cdot \arcsin (5 x) - 20 x \cdot \arcsin (5 x) + {(1 - 25 x^{2})}^{\frac{3}{2}})

Factor identified:

500 x^{3} \cdot \arcsin (5 x) - 20 x \cdot \arcsin (5 x) + {(1 - 25 x^{2})}^{\frac{3}{2}}

▶Derivative:

5 \cdot (500 x^{3} \cdot \arcsin (5 x) - 20 x \cdot \arcsin (5 x) + {(1 - 25 x^{2})}^{\frac{3}{2}})