derivative of a*b^x — step-by-step solution

The expression

a b^{x}

contains several variables: the variable x is chosen for differentiation

Let's differentiate:

\frac{d}{d x} (a b^{x})

To differentiate on intervals where

b > 0

, write

b^{x}

e^{x \cdot \ln (b)}

ⅰ

\frac{d}{d x} (b^{x})

\frac{d}{d x} (x \cdot \ln (b)) \cdot e^{x \cdot \ln (b)}

a \cdot \frac{d}{d x} (x \cdot \ln (b)) \cdot e^{x \cdot \ln (b)}

ⓘ For differentiation with respect to

x

\ln (b)

is treated as constant. A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (x \cdot \ln (b))

\ln (b) \cdot \frac{d}{d x} (x)

\ln (b) \cdot 1

\ln (b)

■As exp and ln are inverse of each other, exp(n*ln(x)) = x^n:

e^{x \cdot \ln (b)}

b^{x}

a \cdot \ln (b) \cdot b^{x}

▶Derivative:

a b^{x} \cdot \ln (b)