Question

In Exercises \(89-98\), find the derivative of the function. \(f(x)=4^{x}\)

Short answer

The derivative of the function \(f(x) = 4^x\) is \(f'(x) = 4^x \cdot ln(4)\)

Step-by-step solution

Identify Function Type

The given function \(f(x) = 4^x\) is an exponential function, where the base a is a constant while the exponent is a variable.

Apply Derivative Formula

For the exponential function \(f(x) = a^x\), the rule for its derivative is \(f'(x) = a^x \cdot ln(a)\). Using this rule, the derivative of our function \(f(x) = 4^x\) yields \(f'(x) = 4^x \cdot ln(4)\)

Simplify

The result calculated in the previous step represents the derivative of the function in its simplest form and cannot be further simplified. Therefore, the derivative of \(f(x) = 4^x\) is \(f'(x) = 4^x \cdot ln(4)\)