f(x)= 5x^2+8x+7/x^1/2
i have to find the derivative and when I do this is what i get....
(x^.5)(10x+8)-(5x^2+8x+7)(-.5x^-3/2)/(x)
it says that i am wrong however
please help
To find the derivative of the given function f(x) = 5x^2 + 8x + 7/x^(1/2), we need to apply the quotient rule. The quotient rule states that if we have a function u(x) divided by another function v(x), and both u(x) and v(x) are differentiable, then the derivative of the quotient is given by:
[f'(x) * v(x) - u(x) * v'(x)] / v(x)^2
Let's go step by step to find the derivative:
Step 1: Simplify the given function
f(x) = (5x^2 + 8x + 7) / x^(1/2)
Step 2: Rewrite the function using fractional exponents
f(x) = (5x^2 + 8x + 7) * x^(-1/2)
Step 3: Now, let u(x) = 5x^2 + 8x + 7 and v(x) = x^(-1/2)
Step 4: Find the derivatives of u(x) and v(x)
u'(x) = d/dx(5x^2 + 8x + 7)
= 10x + 8
v'(x) = d/dx(x^(-1/2))
= (-1/2)x^(-3/2)
Step 5: Apply the quotient rule
[f'(x) * v(x) - u(x) * v'(x)] / v(x)^2
= [(10x + 8) * x^(-1/2) - (5x^2 + 8x + 7) * (-1/2)x^(-3/2)] / (x^(-1/2))^2
= [(10x + 8) * x^(-1/2) + (5x^2 + 8x + 7) * (1/2)x^(-3/2)] / x
Simplifying further, we can combine the terms with the same exponents:
= [(10x + 8 + (5x^2 + 8x + 7)/2) * x^(-1/2)] / x
= [(10x + 8 + (5x^2 + 8x + 7)/2)* x^(-1/2)] / x^1
Combining the denominators:
= [(10x + 8 + (5x^2 + 8x + 7)/2)] / (x^(1/2) * x)
Therefore, after simplification, the derivative is:
f'(x) = (10x + 8 + (5x^2 + 8x + 7)/2) / (x^(3/2))
Please note that it is always a good idea to double-check your work and simplify your final answer.