Derivative of b^x
To find the derivative of b^x, where b is a constant, you can use the concept of logarithmic differentiation
To find the derivative of b^x, where b is a constant, you can use the concept of logarithmic differentiation.
Let y = b^x.
To find dy/dx, take the natural logarithm of both sides:
ln(y) = ln(b^x)
Using the logarithmic property, ln(b^x) = x * ln(b):
ln(y) = x * ln(b)
Now, differentiate both sides with respect to x:
(d/dx) ln(y) = (d/dx) (x * ln(b))
Using the chain rule, the derivative of ln(y) with respect to x is (1/y) * dy/dx:
(1/y) * dy/dx = ln(b)
Multiplying both sides by y:
dy/dx = y * ln(b)
Since y = b^x, substitute it back into the equation:
dy/dx = b^x * ln(b)
So, the derivative of b^x with respect to x is b^x times the natural logarithm of b: dy/dx = b^x * ln(b).
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