Questions: Consider the following function: g(x) = (1+2x^6-5x^7)/x Find the derivative of the function. g'(x) =

Solution Steps

To find the derivative of the function \( g(x) = \frac{1 + 2x^6 - 5x^7}{x} \), we can first simplify the function by dividing each term in the numerator by \( x \). Then, we can apply the power rule for derivatives to each term separately.

The given function is \( g(x) = \frac{1 + 2x^6 - 5x^7}{x} \). We can simplify this by dividing each term in the numerator by \( x \): \[ g(x) = \frac{1}{x} + 2x^5 - 5x^6 \]

To find the derivative \( g'(x) \), we apply the power rule to each term:

• The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).
• The derivative of \( 2x^5 \) is \( 10x^4 \).
• The derivative of \( -5x^6 \) is \( -30x^5 \).

Thus, the derivative is: \[ g'(x) = -\frac{1}{x^2} + 10x^4 - 30x^5 \]

Final Answer