derivative of (x^8-2x^4-10x)/(x^4)

⚠Ⓘ 2x is read as the product 2*x: for a power, write 2^x‖10x is read as the product 10*x: for a power, write 10^x

1) Let's perform the differentiation:

\frac{d}{d x} (\frac{x^{8} - 2 x^{4} - 10 x}{x^{4}})

Let's factorize in

ℝ

x^{8} - 2 x^{4} - 10 x

You can write the expression in factored form:

x \cdot (x^{7} - 2 x^{3} - 10)

The fraction:

\frac{x}{x^{4}}

simplifies by the common factor:

x

into the new fraction:

\frac{1}{x^{3}}

\frac{d}{d x} (\frac{x^{7} - 2 x^{3} - 10}{x^{3}})

We differentiate

u1

x^{7} - 2 x^{3} - 10

and

v2

x^{3}

using:

\frac{d}{d x} (\frac{u1}{v2}) = \frac{\frac{d}{d x} (u1) \cdot v2 - u1 \cdot \frac{d}{d x} (v2)}{{v2}^{2}}

2) Let's differentiate

u1

\frac{d}{d x} (x^{7} - 2 x^{3} - 10)

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

ℹ

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

7 x^{6}

‖

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

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

‖

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

3 x^{2}

7 x^{6} - 2 \cdot 3 x^{2}

7 x^{6} - 6 x^{2}

3) Let's differentiate

v2

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

3 x^{2}

4) Let's evaluate the resulting derivative (

\frac{u1}{v2}

)':

\frac{(7 x^{6} - 6 x^{2}) \cdot x^{3} - (x^{7} - 2 x^{3} - 10) \cdot 3 x^{2}}{{(x^{3})}^{2}}

ℹ

\frac{(7 x^{6} - 6 x^{2}) \cdot x^{3} - (x^{7} - 2 x^{3} - 10) \cdot 3 x^{2}}{{(x^{3})}^{2}}

\frac{x^{2} \cdot (x \cdot (7 x^{6} - 6 x^{2}) - 3 \cdot (x^{7} - 2 x^{3} - 10))}{{(x^{3})}^{2}}

‖

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

x^{(3 \cdot 2)}

\frac{x^{2} \cdot (x \cdot (7 x^{6} - 6 x^{2}) - 3 \cdot (x^{7} - 2 x^{3} - 10))}{x^{(3 \cdot 2)}}

\frac{x^{2} \cdot (x \cdot (7 x^{6} - 6 x^{2}) - 3 \cdot (x^{7} - 2 x^{3} - 10))}{x^{6}}

We divide

\frac{x^{2}}{x^{6}}

by the common factor:

x^{2}

, which gives the new fraction:

\frac{1}{x^{4}}

\frac{x \cdot (7 x^{6} - 6 x^{2}) - 3 \cdot (x^{7} - 2 x^{3} - 10)}{x^{4}}

ℹ

x \cdot (7 x^{6} - 6 x^{2})

x^{7} \cdot 7 + x^{3} \cdot (-6)

‖

- 3 \cdot (x^{7} - 2 x^{3} - 10)

- (3 x^{7} - 6 x^{3} - 30)

\frac{(x^{7} \cdot 7 + x^{3} \cdot (-6)) - (3 x^{7} - 6 x^{3} - 30)}{x^{4}}

\frac{x^{7} \cdot 7 - x^{3} \cdot 6 - 3 x^{7} + 6 x^{3} + 30}{x^{4}}

ℹ

x^{7} \cdot 7 - x^{3} \cdot 6 - 3 x^{7}

4 x^{7} - 6 x^{3}

‖

- 6 x^{3} + 6 x^{3}

0

\frac{4 x^{7} + 0 + 30}{x^{4}}

▶Derivative:

\frac{4 x^{7} + 30}{x^{4}}