d/dx[b^x]=
ln(b)b^x
To differentiate d/dx[b^x], we can use the chain rule. Let y = b^x, then:
d/dx[b^x] = d/dx[y] = dy/dx
Now we need to find dy/dx. We can rewrite y as e^(ln(b^x)) and use the chain rule again:
dy/dx = d/dx[e^(ln(b^x))] = e^(ln(b^x)) * d/dx[ln(b^x)]
Using the chain rule again for d/dx[ln(b^x)] = (1/ln(b)) * d/dx[b^x]:
dy/dx = e^(ln(b^x)) * (1/ln(b)) * d/dx[b^x]
Since e^(ln(b^x)) = b^x, we can substitute that and simplify:
dy/dx = b^x * (1/ln(b)) * d/dx[b^x]
Therefore, the final answer is:
d/dx[b^x] = b^x * (ln(b)) * d/dx[x]
Or:
d/dx[b^x] = b^x * ln(b)
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