d/dx[a^x]=
a^x*ln(a)
To find the derivative of a function of the form a^x with respect to x, we can use the chain rule. Let y = a^x, then we can write:
y = e^(ln(a^x)) (using the fact that a^x = e^(ln(a^x)))
y = e^(x ln(a)) (using the power rule of logarithms)
Now, we can differentiate both sides with respect to x:
dy/dx = d/dx [e^(x ln(a))]
dy/dx = e^(x ln(a)) * d/dx [x ln(a)] (using the chain rule)
dy/dx = e^(x ln(a)) * (ln(a)) (using the derivative of x ln(a))
Finally, substituting y = a^x, we have:
d/dx [a^x] = d/dx [e^(x ln(a))] = a^x * ln(a)
Therefore, the derivative of a^x with respect to x is given by a^x times the natural logarithm of a, i.e., d/dx [a^x] = a^x * ln(a).
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