1) Find the second derivative of the function f(s)= s^4 sec s. I know the answer

ID: 2866050 • Letter: 1

Question

1) Find the second derivative of the function f(s)= s^4 sec s.

I know the answer is f '' s^2 sec s (12+s^2 sec^2 s + 8s tan s+ s^2 tan^2 s), I just want to know how they got the answer.

2) Find the derivative of the function f(x)= x^4 (2+5x)^7.

I know the answer is f '(x)= x^3 (2+5x)^6 (8+55x), I just want to know how they got the answer.

3) Find the second derivative of the function f(x)= (2x^4 +8)^4.

I know the answer is f ''(x)= 32x^2 (8+2x^4)^2 (24+30x^4), I just want to know how they got the answer.

1) f(s)= s^4 sec s.

f '(s) = [4s^3 secs ]+ [s^4 secs tans]

f "(s) = (12s^2 secs) + (4s^3 secstanx) + ( 4s^3 secs tans)+ (s^4 secs tans tans) + (s^4 secs sec^2 s)

=> f "(s) = (12s^2 secs) + (4s^3 secstanx) + (4s^3 secs tans) + (s^4 secs tan^2s) + (s^4 sec^3s)

=> f"(s) = (12s^2 secs) + (8s^3 secstanx) + (s^4 secs tan^2s) + (s^4 sec^3 s)

= s^2 secs ( 12 + 8s secs tans + s^2tan^2s+ s^2sec^2s)

2) f(x)= x^4 (2+5x)^7.

=> f '(x) = [4x^3 (2+5x)^7] + [ x^4 * 7(2+5x)^6 * 5]

= x^3 (2+5x)^6[ 4(2+5x) + 35x]

=x^3 (2+5x)^6[ 8 + 55x)

3) f(x)= (2x^4 +8)^4.

=> f'(x) = 4 (2x^4 +8)^3 * d/dx[ (2x^4+8)

= 4 (2x^4 +8)^3 * 8x^3

= 32x^3 (2x^4 +8)^3

f "(x) = [96x^2(2x^4+8)^3 ] + [ 32x^3 * 3(2x^4+8)^2 * 8x^3]

=32x^2 (2x^4+8)^2 [ 3(2x^4+8) + 24x^4]

= 32x^2 (2x^4+8)^2 ( 30x^4 + 24)

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