How do you find the derivative when the variable is in the exponent?!
For example, what's the derivative of 2^x ?
For example, what's the derivative of 2^x ?
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essentially you have
y = 2^x and you want to find dy/dx
take the ln of both sides
ln y = x ln2
take the derivative
1/y (dy/dx) = ln2
and dy/dx = y ln2 = 2^x ( ln2)
y = 2^x and you want to find dy/dx
take the ln of both sides
ln y = x ln2
take the derivative
1/y (dy/dx) = ln2
and dy/dx = y ln2 = 2^x ( ln2)
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If you don't have the formula memorized, you can differentiate logarithmically.
y = 2^x
ln y = x ln 2
1/y dy/dx = 1/x ln 2
dy/dx = y/x ln 2
dy/dx = 2^x/x ln 2 = 2^x ln 2/x
y = 2^x
ln y = x ln 2
1/y dy/dx = 1/x ln 2
dy/dx = y/x ln 2
dy/dx = 2^x/x ln 2 = 2^x ln 2/x
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d/dx(a^x) = lna(a^x) so
d/dx(2^x) = ln2(2^x)
d/dx(2^x) = ln2(2^x)