Question

If f(x) = e^xg(x), g(0) = 2, g'(0) = 1, then f'(0) = __________________.

Answer 1

$f\left(x\right)=e^{x}g\left(x\right)$

Differentiating both sides with respect to x, we get

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left[e^{x}g\left(x\right)\right]$

$\Rightarrow f\text{'}\left(x\right)=e^x\times \frac{d}{dx}g\left(x\right)+g\left(x\right)\times \frac{d}{dx}{e}^x$

$\Rightarrow f\text{'}\left(x\right)=e^x\times g\text{'}\left(x\right)+g\left(x\right)\times e^x$

Putting x = 0, we get

$f\text{'}\left(0\right)=e^0\times g\text{'}\left(0\right)+g\left(0\right)\times e^0$

$\Rightarrow f\text{'}\left(0\right)=1\times 1+2\times 1$ [g(0) = 2, g'(0) = 1 and e⁰ = 1]

$\Rightarrow f\text{'}\left(0\right)=1+2=3$

Thus, the value of $f\text{'}\left(0\right)$ is 3.


If f(x) = e^xg(x), g(0) = 2, g'(0) = 1, then f'(0) = ____3____.