Twice derived

Calculus Level 2

Let f ( x ) = ( x 2 × e x ) f(x) = (x^2 \times e^x) .

Find f ′ ′ ( 2 ) f''(2) to 2 decimal places.


The answer is 103.45.

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Sep 9, 2016

By product rule:

f ′ ( x ) = ( ( x 2 ) ′ × e x ) + ( x 2 × ( e x ) ′ ) = ( 2 x × e x ) + ( x 2 × e x ) f'(x) = ((x^2)' \times e^x) + (x^2 \times (e^x)') = (2x \times e^x) + (x^2 \times e^x)

so by double application of product rule:

f ′ ′ ( x ) = ( ( ( 2 x ) ′ × e x ) + ( 2 x × ( e x ) ′ ) + ( ( ( x 2 ) ′ × e x ) + ( x 2 × ( e x ) ′ ) ) f''(x) = (((2x)' \times e^x) + (2x \times (e^x)') + (((x^2)' \times e^x) + (x^2 \times (e^x)'))

This simplifies to ( x 2 + 4 x + 2 ) × ( e x ) (x^2 + 4x + 2) \times (e^x)

Substituting x = 2 x = 2 yields 14 e 2 = 103.45 14e^2 = 103.45

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