derivative of (3x-2)^3(8x^2+7)^3

1) Let's differentiate:

\frac{d}{d x} ({(3 x - 2)}^{3} \cdot {(8 x^{2} + 7)}^{3})

For

u1

{(3 x - 2)}^{3}

and

v2

{(8 x^{2} + 7)}^{3}

, 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 perform the differentiation

u1

\frac{d}{d x} ({(3 x - 2)}^{3})

Apply the differentiation rule:

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

with f =

3 x - 2

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(3 x - 2)}^{3})

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

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

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

\frac{d}{d x} (3 x - 2)

\frac{d}{d x} (3 x) + 0

3 \cdot (\frac{d}{d x} (3 x) + 0) \cdot {(3 x - 2)}^{2}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (3 x) + 0

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

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

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

3

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

ⅰ

3 \cdot 3

9

9 \cdot {(3 x - 2)}^{2}

3) Let's differentiate

v2

\frac{d}{d x} ({(8 x^{2} + 7)}^{3})

We differentiate using:

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

(where f =

8 x^{2} + 7

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(8 x^{2} + 7)}^{3})

3 \cdot \frac{d}{d x} (8 x^{2} + 7) \cdot {(8 x^{2} + 7)}^{2}

3 \cdot \frac{d}{d x} (8 x^{2} + 7) \cdot {(8 x^{2} + 7)}^{2}

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

\frac{d}{d x} (8 x^{2} + 7)

\frac{d}{d x} (8 x^{2}) + 0

3 \cdot (\frac{d}{d x} (8 x^{2}) + 0) \cdot {(8 x^{2} + 7)}^{2}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (8 x^{2}) + 0

\frac{d}{d x} (8 x^{2})

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

\frac{d}{d x} (8 x^{2})

8 \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

3 \cdot 8 \cdot 2 x \cdot {(8 x^{2} + 7)}^{2}

ⅰ

3 \cdot 8

24

24 \cdot 2 x \cdot {(8 x^{2} + 7)}^{2}

ⅰ

24 \cdot 2

(20 + 4) \cdot 2

20 \cdot 2 + 4 \cdot 2

40 + 8

48

48 x \cdot {(8 x^{2} + 7)}^{2}

4) Then you can resume the differentiation:

ⅰ

\frac{d}{d x} ({(3 x - 2)}^{3} \cdot {(8 x^{2} + 7)}^{3})

9 \cdot {(3 x - 2)}^{2} \cdot {(8 x^{2} + 7)}^{3} + {(3 x - 2)}^{3} \cdot 48 x \cdot {(8 x^{2} + 7)}^{2}

9 \cdot {(3 x - 2)}^{2} \cdot {(8 x^{2} + 7)}^{3} + {(3 x - 2)}^{3} \cdot 48 x \cdot {(8 x^{2} + 7)}^{2}

48 x \cdot {(3 x - 2)}^{3} \cdot {(8 x^{2} + 7)}^{2} + 9 \cdot {(3 x - 2)}^{2} \cdot {(8 x^{2} + 7)}^{3}

You can write the expression in factored form:

{(3 x - 2)}^{2} \cdot {((8 x^{2} + 7))}^{2} \cdot (48 x \cdot (3 x - 2) + 9 \cdot (8 x^{2} + 7))

Let's factorize in

ℝ

(48 x \cdot (3 x - 2) + 9 \cdot (8 x^{2} + 7))

We factor out

3

, the greatest common divisor of the coefficients:

48 x \cdot (3 x - 2) + 9 \cdot (8 x^{2} + 7)

3 \cdot (16 x \cdot (3 x - 2) + 3 \cdot (8 x^{2} + 7))

Factor found:

16 x \cdot (3 x - 2) + 3 \cdot (8 x^{2} + 7)

▶Derivative:

3 \cdot {(3 x - 2)}^{2} \cdot {(8 x^{2} + 7)}^{2} \cdot (16 x \cdot (3 x - 2) + 3 \cdot (8 x^{2} + 7))