Question
Differentiate the following functions from first principles:
eax+b.
eax+b.
⇒ f(x + h) = ea(x+h)+b
$\therefore\frac{\text{d}}{\text{dx}}(\text{f(x)})=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^{\text{a}(\text{x}+\text{h})+\text{b}}-\text{e}^{(\text{ax}+\text{b})}}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^{\text{ax}+\text{b}}\text{e}^{\text{ah}}-\text{e}^{\text{ax}+\text{b}}}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\text{ e}^{\text{ax}+\text{b}}\left\{\frac{(\text{e}^{\text{ah}}-1)}{\text{ah}}\right\}\times\text{a}$ $=\text{ae}^{\text{ax}+\text{b}} \lim\limits_{\text{h}\rightarrow0}\left\{\frac{(\text{e}^{\text{ah}}-1)}{\text{ah}}\right\}$ $=\text{ae}^{\text{ax}+\text{b}}$ So, $\frac{\text{d}}{\text{dx}}(\text{e}^{\text{ax}}+\text{b})=\text{ae}^{\text{ax}+\text{b}}$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\int^{\frac{\pi}{4}}_{0}\frac{\sin\text{x}+\cos\text{x}}{9+16\sin2\text{x}}\text{dx}$
| Bag | Colour of the ball | ||
| Black | White | Red | |
| I | 1 | 2 | 3 |
| II | 2 | 4 | 1 |
| III | 4 | 5 | 3 |