Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the derivative of a^x?

1. ln(a) a^x

2. (a^x)ln a

3. (a^x)/x

4. ln(x) a^x

The correct answer is: (a^x)ln a

The derivative of the function \( a^x \), where \( a \) is a constant, can be derived using the properties of exponential functions and logarithms. When differentiating \( a^x \), it is helpful to express it in terms of the natural exponential function: \[ a^x = e^{x \ln(a)} \] By applying the chain rule to the derivative of \( e^{u(x)} \) where \( u(x) = x \ln(a) \), we find: \[ \frac{d}{dx}(e^{u(x)}) = e^{u(x)} \cdot u'(x) \] Calculating \( u'(x) \), we observe that: \[ u'(x) = \ln(a) \] Now, substituting back into the differentiation formula gives us: \[ \frac{d}{dx}(a^x) = e^{x \ln(a)} \cdot \ln(a) = \ln(a) \cdot e^{x \ln(a)} = \ln(a) \cdot a^x \] Thus, the derivative of \( a^x \) is \( \ln(a) \cdot a