How do you find the derivative of (x+1) times the third root of (x-5)spuared(x=4)?
Do you mean
f9x) = (x+1)[(x-5)^2*(x-4)]^1/3 ?
Which terms are included in the cube root?
The x=4 in parentheses makes no sense
Yes the eqaution you wrote is correct. There is no f9x however.
I meant f(x). It represent the "function of x" that you wrote.
Rewrite your function as
(x+1)*(x-5)^(2/3)*(x-4)^1/3
Use the rule that
d/dx (f*g*h) = f d/dx(g*h)+ gh df/dx
= fg dh/dx + fh dg/dx + gh df/dx
where f, g and h are functions of x.
That makes the derivative
(x-5)^(2/3)*(x-4)^1/3
+ (2/3)(x-4)(-1/3)*(x-1)(x-4)^(1/3)
+ (x+1)*(x-5)^(2/3)*(1/3)(x-4)^(-2/3)
Hey thanks a lot.
To find the derivative of the given function at x = 4, you can follow these steps:
Step 1: Simplify the function.
The given function is (x+1) * ((x-5)^(1/3))^2.
Step 2: Apply the product rule.
To differentiate a product of two functions, we use the product rule. The product rule states that if we have two functions, f(x) and g(x), the derivative of their product, f(x) * g(x), is given by:
(f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x),
where f'(x) represents the derivative of f(x) and g'(x) represents the derivative of g(x).
Step 3: Find the derivatives of each part.
Let's differentiate (x+1) and ((x-5)^(1/3))^2 separately.
The derivative of (x+1) is 1.
To differentiate ((x-5)^(1/3))^2, we can apply the chain rule:
Let u = (x-5)^(1/3). Then u^2 = ((x-5)^(1/3))^2.
We can find the derivative of u^2 using the chain rule:
(d(u^2) / dx) = 2u * (du/dx).
To find (du/dx), we differentiate u = (x-5)^(1/3) using the power rule:
(du/dx) = (1/3) * (x-5)^(-2/3).
Step 4: Substitute the values and calculate.
Now, substitute the values into the derivative formula and calculate:
(f(x) * g(x))' = 1 * ((x-5)^(1/3))^2 + (x+1) * 2 * ((1/3) * (x-5)^(-2/3)).
Evaluate the expression at x = 4 to find the derivative at x = 4.