Derivative of b^x
b^x ln(b)
The derivative of b^x is given by:
d/dx[b^x] = ln(b)*b^x
Where ln(b) is the natural logarithm of b.
To understand how this formula is derived, we can use the definition of a derivative.
Let y = b^x.
Taking logarithm on both sides, we get:
ln(y) = ln(b^x) = x*ln(b)
Now, we can differentiate both sides of the equation with respect to x using the chain rule:
d/dx[ln(y)] = d/dx[x*ln(b)]
1/y * dy/dx = ln(b)
dy/dx = y*ln(b)
Substituting y = b^x gives:
d/dx[b^x] = ln(b)*b^x
Therefore, the derivative of b^x is proportional to the value of b^x itself, with ln(b) serving as the proportionality constant.
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