Final answer:
To differentiate h(x) = e^{x^9 + ln(x)}, we used the chain rule and the properties of exponents and logarithms. After simplifying, the derivative is h'(x) = e^{x^9} + 9x^8 ⋅ e^{x^9}, correcting the initial mistaken expression provided in the question.
Step-by-step explanation:
The student is asking how to differentiate the function h(x) = e^{x^9 + ln(x)}. To differentiate this function, we use the chain rule and the properties of exponents and logarithms. The function can be rewritten as h(x) = e^{x^9} ⋅ e^{ln(x)}. Since e and ln are inverse functions, e^{ln(x)} = x. Therefore, we simplify h(x) to h(x) = x ⋅ e^{x^9}. Differentiating h(x) with respect to x gives h'(x) = e^{x^9} + 9x^8 ⋅ e^{x^9}, as the derivative of x is 1 and the derivative of e^{x^9} is 9x^8 ⋅ e^{x^9}, using the chain rule. Thus, the correct differentiation is h'(x) = e^{x^9} + 9x^8 ⋅ e^{x^9}, not h'(x) = 9e^{x^9} ⋅ x^8 + 1/x as stated in the question.