How do you find the derivative of e^xln2?

dy/dx
= de^(xln2)/d(xln2) * d(xln2)/dx (chain rule)
= e^(xln2) * ln2
= ln2 * [e^(ln2)]^x
= ln2 * 2^x

Given:

e^{(x)ln(2)}

Use the identity (a)ln(b) = ln(b^a)

e^ln(2^x)

Use the property e^ln(u) = u:

2^x

Let y = 2^x so that one can use logarithmic differentiation:

Take the natural log (ln) of both sides:

ln(y) = ln(2^x)

ln(y) = (x)ln(2)

differentiate both sides with respect to x:

(1/y)(dy/dx) = ln(2)

Multiply by y:

dy/dx = ln(2)y

Substitute 2^x for y:

dy/dx = ln(2)2^x