derivative of f(x)=(x^2+3)^3(x^3+3)^2

We save the definition:

f (x) = {(x^{2} + 3)}^{3} \cdot {(x^{3} + 3)}^{2}

1) Let's perform the differentiation:

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

We differentiate

u1

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

and

v2

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

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} + 3)}^{3})

Apply the differentiation rule:

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

with f =

x^{2} + 3

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

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

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

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

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

ⓘ Adding 0 changes nothing:

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

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

ⅰ

3 \cdot 2

6

6 x \cdot {(x^{2} + 3)}^{2}

3) Let's differentiate

v2

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

We differentiate using:

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

(where f =

x^{3} + 3

)

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Expand by multiplying each term inside the parentheses:

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

2 \cdot \frac{d}{d x} (x^{3} + 3) \cdot x^{3} + 6 \cdot \frac{d}{d x} (x^{3} + 3)

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

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

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

2 \cdot (\frac{d}{d x} (x^{3}) + 0) \cdot x^{3} + 6 \cdot (\frac{d}{d x} (x^{3}) + 0)

ⓘ Adding 0 changes nothing:

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

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

■The derivative of x^n is n*x^(n-1):

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

3 x^{2}

2 \cdot 3 x^{2} \cdot x^{3} + 6 \cdot 3 x^{2}

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 3

6

6 \cdot 3

18

■Multiplying powers with the same base adds the exponents:

x^{2} x^{3}

x^{5}

6 x^{5} + 18 x^{2}

4) You can then resume the differentiation:

ⅰ

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

6 x \cdot {(x^{2} + 3)}^{2} \cdot {(x^{3} + 3)}^{2} + {(x^{2} + 3)}^{3} \cdot (6 x^{5} + 18 x^{2})

6 x \cdot {(x^{2} + 3)}^{2} \cdot {(x^{3} + 3)}^{2} + {(x^{2} + 3)}^{3} \cdot (6 x^{5} + 18 x^{2})

{(x^{2} + 3)}^{2} \cdot (6 x \cdot {(x^{3} + 3)}^{2} + (x^{2} + 3) \cdot (6 x^{5} + 18 x^{2}))

ⓘ A square being never negative,

x^{2} + 3

is never zero and cannot be factored in ℝ

▶Derivative:

{(x^{2} + 3)}^{2} \cdot (6 x \cdot {(x^{3} + 3)}^{2} + (x^{2} + 3) \cdot (6 x^{5} + 18 x^{2}))