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Q.
If the two curves $y=a^x$ and $y=b^x$ intersect at an angle $\alpha$, then tan $\alpha$ equals.
Application of Derivatives
Solution:
For the curve $y=a^{x}, \frac{dy}{dx}=a^{x}\,log\,a$
For the curve $y=b^{x}, \frac{dy}{dx}=b^{x}\,log\,b$
The given curves meet at $\left(1, 1\right)$
$\left[\because a^{x}=b^{x} \Rightarrow x=0\right]$
At $\left(0, 1\right)$
$m_{1}=\frac{dy}{dx}$ for first curve $=log\,a$
$m_{2}=\frac{dy}{dx}$ for first curve $=log\,b$
$\therefore tan\,\alpha=\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}$$=\frac{log\,a-log\,b}{1+log\,a\,log\,b}$