Multiple rules needed

Calculus Level 4

Let f ( x ) = ( ( x 2 × ln ⁡ x ) + ( ( cos ⁡ ( x 2 ) ) / ( x + 1 ) ) ) f(x) = ((x^2 \times \ln x) + ((\cos(x^2))/(x+1)))

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


The answer is 5.85.

Denton Young by Denton Young , 47, USA

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2 solutions

Denton Young
Sep 8, 2016

Break the problem into pieces.

f ′ ( x 2 × l n x ) = ( ( 2 x × l n x ) + x ) f'(x^2 \times ln x) = ((2x \times ln x) + x)

f ′ ( ( c o s ( x 2 ) ) / ( x + 1 ) ) = ( ( ( x + 1 ) ( − 2 x s i n ( x 2 ) ) − c o s ( x 2 ) ) / ( ( x + 1 ) 2 ) ) f'((cos(x^2))/(x+1)) = (((x+1)(-2x sin (x^2)) - cos(x^2))/((x+1)^2))

substituting x = 2 and combining: f ′ ( 2 ) = ( ( 4 l n 2 ) + 2 ) + ( ( ( 3 ) ( − 4 s i n 4 ) − c o s 4 ) / 3 2 ) f'(2) = ((4 ln 2) + 2) + (((3)(-4 sin 4) - cos 4)/3^2) = 5.85

1 Helpful 0 Interesting 0 Brilliant 0 Confused
John Crocker
Sep 8, 2016

You should specify in the problem statement what units the trig functions are in.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Everyone who does these knows that trig units are automatically assumed to be in radians unless specified otherwise.

Denton Young - 4 years, 9 months ago

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