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The derivative of y = 3^x is y' = 3^x ln(3).

To differentiate y = 3^x, we use the chain rule. Let u = 3^x, then y = f(u) = u. We have:

y' = f'(u) * u'

To find f'(u), we use the power rule:

f'(u) = d/dx (3^x) = ln(3) * 3^x

To find u', we use the derivative of exponential functions:

u' = d/dx (3^x) = ln(3) * 3^x

Substituting these values into the chain rule formula, we get:

y' = f'(u) * u' = ln(3) * 3^x * 3^x = 3^x ln(3)

Therefore, the derivative of y = 3^x is y' = 3^x ln(3). This means that the slope of the tangent line to the graph of y = 3^x at any point (x, y) is given by 3^x ln(3). The function y = 3^x is an example of an exponential function, which grows very quickly as x increases. The derivative tells us how fast the function is growing at any point, and the value of ln(3) tells us the rate of growth relative to the base of the exponential function.

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