Final answer:
To find the derivative of the given expression 8x⁷ + 10√x + 9/x⁵, we use the power rule and the chain rule. The derivative is 56x⁶ + 5/√x - 45/x⁶.
Step-by-step explanation:
To find the derivative of the given expression, we will use the power rule and the chain rule. Let's break down the expression:
8x⁷ + 10√x + 9/x⁵
For the first term, 8x⁷, we apply the power rule: the derivative of xⁿ is n*xⁿ⁻¹. Therefore, the derivative of 8x⁷ is 56x⁶.
For the second term, 10√x, we use the chain rule. Let u = √x, so the derivative of u is 1/(2√x) = 1/(2u). Applying the chain rule, the derivative of 10u is 10*(1/(2u)) = 5/u = 5/√x.
For the third term, 9/x⁵, we use the power rule again. The derivative of x⁻ⁿ is -n*x⁻ⁿ⁺¹. Therefore, the derivative of 9/x⁵ is -45/x⁶.
Combining these derivatives, the derivative of the entire expression is 56x⁶ + 5/√x - 45/x⁶.