Question

If $f (1)=1, f' (1)=3$ then the derivative of $f ( f ( f (x)))+(f(x))^{2}$ at $x=1$ is

• A. 10
• B. 35
• C. 33
• D. 12

Answer 1

Answer: (C)

$f (1)=1, f ^{\prime}(1)=3$
$\frac{ d }{ dx }\left[ f ( f ( f ( x )))+( f ( x ))^{2}\right]$
${\left[ f ^{\prime}( f ( f ( x ))) \cdot f ^{\prime} f ( x ) \cdot f ^{\prime}( x )+2 f ( x ) \cdot f ^{\prime}( x )\right] }$
$= f ^{\prime}( f ( f (1))) f ^{\prime}( f (1)) \cdot f ^{\prime}(1)+2 f (1) \cdot f ^{\prime}(1)$
$f ^{\prime}( f (1)) f ^{\prime}(1) \cdot 3+2 \cdot(1) 3$
$= f ^{\prime}(1) \cdot 3 \cdot 3+6$
$=27+6=33$