Mastering Derivatives: The Formula And Definition For Deriving The Derivative Of B^X

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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