(a) Find f'(x) if
f(x) = ((2x ^ 2 - 1)(x ^ 2 + 3sqrt(x)))/(x ^ 3 + 2sqrt(x))
f(x) = ((2x^2 - 1)(x^2 + 3√x))/(x^3 + 2√x)
consider this as f(x) = u/v where u and v are functions of x. Then, using the quotient rule,
f'(x) = (u'v - uv')/v^2
Now, u' can be found using the product rule, so we have
f'(x) = (((4x)(x^2 + 3√x))+(3x^2-1)(2x + 3/(2√x)))(x^2 + 3√x) - ((2x^2 - 1)(x^2 + 3√x))(3x^2 + 1/√x)) / (x^3 + 2√x)^2
Now simplify that! You should get something like
x[4x^5+2x^3+48 + √x (28x^2+15x-6)]
-------------------------------------------------
2(x^3 + 2√x)^2
That is very, very difficult to solve and write on this site.
In google type:
online derivative calculator with steps
When you see list of results click on:
Online Derivative Calculator with Steps - eMathHelp
When page be open in rectangle Function paste:
((2x ^ 2 - 1)(x ^ 2 + 3sqrt(x)))/(x ^ 3 + 2sqrt(x))
then click CALCULATE
you will see solution step-by-step
To find the derivative of the function f(x), we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's use the quotient rule to find the derivative of f(x):
First, let's find g'(x) and h'(x):
g(x) = (2x^2 - 1)(x^2 + 3√(x))
g'(x) = (d/dx)(2x^2 - 1)(x^2 + 3√(x))
To find g'(x), we need to use the product rule:
(g'(x) = (d/dx)(2x^2 - 1)(x^2 + 3√(x))
= (2x^2 - 1)(d/dx)(x^2 + 3√(x)) + (x^2 + 3√(x))(d/dx)(2x^2 - 1)
To find (d/dx)(x^2 + 3√(x)), we can differentiate each term separately:
(d/dx)(x^2) = 2x
(d/dx)(3√(x)) = (3/2)(1/√(x)) = (3/2)(1/√(x)) = (3/2)(1/(2√(x))) = (3/2)(1/(2x^(1/2))) = (3/2)(1/(2x^(1/2))) = (3/4)x^(-1/2) = (3/4)/√(x)
Now, let's substitute these values back into g'(x):
g'(x) = (2x^2 - 1)(2x) + (x^2 + 3√(x))(3/4)/√(x)
= 2x(2x^2 - 1) + (3/4)(x^2 + 3√(x))/(√(x))
= 4x^3 - 2x + (3/4)(x^(3/2) + 3x)/(√(x))
Next, let's find h(x) and h'(x):
h(x) = x^3 + 2√(x)
h'(x) = (d/dx)(x^3 + 2√(x))
= (d/dx)(x^3) + (d/dx)(2√(x))
= 3x^2 + (d/dx)(2√(x))
= 3x^2 + 1/√(x)
= 3x^2 + 1/x^(1/2)
=3x^2 + x^(-1/2)
Now that we have g'(x) and h'(x), we can apply the quotient rule:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= ((4x^3 - 2x + (3/4)(x^(3/2) + 3x)/(√(x))) * (x^3 + 2√(x)) - ((2x^2 - 1)(x^2 + 3√(x))) * (3x^2 + x^(-1/2))) / ((x^3 + 2√(x))^2)
Simplifying this expression is a bit complicated due to the presence of fractional powers and radicals. However, at this point, we have successfully found the derivative of f(x) using the quotient rule.
To find the derivative, f'(x), of the function f(x), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x)/h(x), where both g(x) and h(x) are differentiable functions, then the derivative f'(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x) ^ 2)
Let's apply the quotient rule and find f'(x) step by step.
First, let's find the derivative of the numerator, g(x), which is ((2x^2 - 1)(x^2 + 3√x)):
g'(x) = (2x^2 - 1)' * (x^2 + 3√x) + (2x^2 - 1) * (x^2 + 3√x)'
To differentiate (2x^2 - 1), we use the power rule for derivatives, which states that the derivative of x^n is given by nx^(n-1):
(2x^2 - 1)' = 4x
To differentiate (x^2 + 3√x), we can break it down into two parts: x^2 and 3√x. We can use the power rule for derivatives for x^2, and the chain rule for differentiating 3√x.
(x^2 + 3√x)' = (x^2)' + (3√x)'
To differentiate x^2, we use the power rule:
(x^2)' = 2x
To differentiate 3√x, we use the chain rule:
(3√x)' = 3 * (1/2) * x^(-1/2) = 3/2√x
Now, let's find the derivative of the denominator, h(x), which is x^3 + 2√x:
h'(x) = (x^3 + 2√x)'
To differentiate x^3, we again use the power rule:
(x^3)' = 3x^2
To differentiate 2√x, we use the chain rule:
(2√x)' = 2 * (1/2) * x^(-1/2) = 1/√x
Now that we have calculated all the derivatives, we can substitute them back into the quotient rule formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x) ^ 2)
f'(x) = ((4x)(x^2 + 3√x) - ((2x^2 - 1)(x^2 + 3√x) * (3/2√x))) / ((x^3 + 2√x) ^ 2)
Simplifying further, we get:
f'(x) = (4x(x^2 + 3√x) - (3/2√x)(2x^2 - 1)(x^2 + 3√x)) / ((x^3 + 2√x) ^ 2)
This is the derivative of the function f(x).