Questions: Find the derivative of f(x) = 4^x(5x^3 + 4/x^2)

Solution Steps

To find the derivative of the function \( f(x) = 4^x \left(5x^3 + \frac{4}{x^2}\right) \), we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions \( u(x) \) and \( v(x) \) is given by \( u'(x)v(x) + u(x)v'(x) \). Here, \( u(x) = 4^x \) and \( v(x) = 5x^3 + \frac{4}{x^2} \). We will also need to use the chain rule to differentiate \( 4^x \).

We start with the function \( f(x) = 4^x \left(5x^3 + \frac{4}{x^2}\right) \). Here, we identify two components:

• \( u(x) = 4^x \)
• \( v(x) = 5x^3 + \frac{4}{x^2} \)

Next, we compute the derivatives of \( u \) and \( v \):

• The derivative of \( u \) is given by: \[ u' = 4^x \log(4) \]
• The derivative of \( v \) is: \[ v' = 15x^2 - \frac{8}{x^3} \]

Using the product rule, we find the derivative of \( f(x) \): \[ f'(x) = u'v + uv' \] Substituting the derivatives we computed: \[ f'(x) = 4^x \log(4) \left(5x^3 + \frac{4}{x^2}\right) + 4^x \left(15x^2 - \frac{8}{x^3}\right) \]

We can factor out \( 4^x \) from the expression: \[ f'(x) = 4^x \left( \left(15x^2 - \frac{8}{x^3}\right) + \log(4) \left(5x^3 + \frac{4}{x^2}\right) \right) \] This simplifies to: \[ f'(x) = 4^x \left( 15x^5 + x \left(5x^5 + 4\right) \log(4) - \frac{8}{x^3} \right) \]

Final Answer