What is the derivative of 2 + 6 x + 5 x 2 + 7 x 3 x 2 ?

Question

What is the derivative of             2      +      6      x      +      5              x                  2                    +      7              x                  3                            x              2            ?

girni67v · Accepted Answer

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{g (x)^{2}}

where

f (x) = 2 + 6 x + 5 x^{2} + 7 x^{3}

and

g (x) = x^{2}

\frac{x^{2} \frac{d}{d x} [2 + 6 x + 5 x^{2} + 7 x^{3}] - (2 + 6 x + 5 x^{2} + 7 x^{3}) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Differentiate.

\frac{x^{2} (0 + \frac{d}{d x} [6 x] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [7 x^{3}]) - (2 + 6 x + 5 x^{2} + 7 x^{3}) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Evaluate

\frac{d}{d x} [6 x]

\frac{x^{2} (0 + 6 + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [7 x^{3}]) - (2 + 6 x + 5 x^{2} + 7 x^{3}) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Evaluate

\frac{d}{d x} [5 x^{2}]

\frac{x^{2} (0 + 6 + 10 x + \frac{d}{d x} [7 x^{3}]) - (2 + 6 x + 5 x^{2} + 7 x^{3}) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Evaluate

\frac{d}{d x} [7 x^{3}]

\frac{x^{2} (0 + 6 + 10 x + 21 x^{2}) - (2 + 6 x + 5 x^{2} + 7 x^{3}) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Simplify

\frac{x^{2} (21 x^{2} + 10 x + 6) - (2 + 6 x + 5 x^{2} + 7 x^{3}) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

is

n x^{n - 1}

where n=2.

\frac{x^{2} (21 x^{2} + 10 x + 6) - (2 + 6 x + 5 x^{2} + 7 x^{3}) (2 x)}{(x^{2})^{2}}

Simplify.

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

What is the derivative of 2 + 6 x + 5 x

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