Question

let ( f ) and ( g ) be differentiable functions such that ( f(0)=3 ) and ( g(0)=7 ). if ( h(x)=3f(x)-2g(x)-5cos x - 3 ), what is the value of ( h(0) )?

Explanation:

Step1: Apply sum - difference rule of derivatives

The sum - difference rule states that if $h(x)=3f(x)-2g(x)-5\cos x - 3$, then $h'(x)=3f'(x)-2g'(x)+5\sin x$.

Step2: Evaluate $h'(x)$ at $x = 0$

Substitute $x = 0$ into $h'(x)$. We know that $f'(0)=3$ and $g'(0)=7$. Also, $\sin(0)=0$. So $h'(0)=3f'(0)-2g'(0)+5\sin(0)$.

Step3: Calculate the value

$h'(0)=3\times3-2\times7 + 5\times0=9 - 14+0=- 5$.

Answer:

$-5$