Derivative of b^x
The derivative of b^x, where b is a constant, can be found using logarithmic differentiation
The derivative of b^x, where b is a constant, can be found using logarithmic differentiation.
Let’s start by using the property of logarithms:
b^x = e^(ln(b^x)) = e^(x * ln(b))
Now, we can differentiate both sides of the equation with respect to x.
d/dx (b^x) = d/dx (e^(x * ln(b)))
To find the derivative of e^(x * ln(b)), we can use the chain rule. Let u = x * ln(b), then:
d/dx (e^(x * ln(b))) = d/du (e^u) * du/dx = e^u * ln(b)
Substituting back in the value of u:
= e^(x * ln(b)) * ln(b) = b^x * ln(b)
Therefore, the derivative of b^x is b^x * ln(b).
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