Find (dy)/(dx) given y=a^x+b^x . where a and b are constants.

To find $\frac{dy}{dx}$ for $y = a^x + b^x$, where $a$ and $b$ are constants, we need to differentiate $y$ with respect to $x$.<br /><br />Given:<br />\[ y = a^x + b^x \]<br /><br />We will use the chain rule for differentiation. The chain rule states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.<br /><br />First, let's differentiate each term separately.<br /><br />1. Differentiate $a^x$ with respect to $x$:<br />\[ \frac{d}{dx}(a^x) = a^x \ln(a) \]<br /><br />2. Differentiate $b^x$ with respect to $x$:<br />\[ \frac{d}{dx}(b^x) = b^x \ln(b) \]<br /><br />Now, combine these results:<br />\[ \frac{dy}{dx} = \frac{d}{dx}(a^x) + \frac{d}{dx}(b^x) \]<br />\[ \frac{dy}{dx} = a^x \ln(a) + b^x \ln(b) \]<br /><br />Therefore, the derivative of $y = a^x + b^x$ with respect to $x$ is:<br />\[ \frac{dy}{dx} = a^x \ln(a) + b^x \ln(b) \]