Find the derivative of the function.

1. F(x)= (1+2x+x^3)^1/4

2. f(t)= (1+tant)^1/3

1.

1 turns to 0
2x turns to 2
x^3 turns to 3x^2 (bring 3 up front & subtract 1 from exponent)
^1/4 goes up front and subtract 1 from that =-3/4

Final answer would be

1/4(3x^2+2)^-3/4

2. Use same procedure
1/3 (sec^2t) ^ -2/3

d/dx (u^n) = n u^(n-1) du/dx

Not quite Will.
I get
(1/4)(x^3+2x+2)^-3/4 [3x^2+2]

f(t)= (1+tant)^1/3

f'(t) = (1/3)(1+tan t)^-(2/3) [ sec^2 t]

Damon is right, I forgot to apply the chain rule.

To find the derivative of a function, you can use the power rule and chain rule, if necessary. Let's find the derivatives of the given functions:

1. F(x) = (1 + 2x + x^3)^(1/4)

To find the derivative of F(x), we will use the chain rule because we have a composition of functions (power and polynomial functions).

Step 1: Apply the power rule to the outer function:

Derivative of u^n = n*u^(n-1)

Where u refers to the function inside the parentheses and n is the power.

Applying the power rule, we get:

d/dx (u^(1/4)) = (1/4) * u^(-3/4) * du/dx

Step 2: Determine the derivative of the inner function u = (1 + 2x + x^3):

The derivative of u with respect to x is:

du/dx = d/dx [ (1 + 2x + x^3) ] = 2 + 3x^2

Step 3: Substitute the values back into the chain rule formula:

d/dx (F(x)) = (1/4) * (1 + 2x + x^3)^(-3/4) * (2 + 3x^2)

Therefore, the derivative of F(x) is:

d/dx (F(x)) = (1/4) * (1 + 2x + x^3)^(-3/4) * (2 + 3x^2)

2. f(t) = (1 + tan(t))^1/3

To find the derivative of f(t), we will again use the chain rule because we have a composition of functions (power and tangent functions).

Step 1: Apply the power rule to the outer function:

Derivative of u^n = n*u^(n-1)

Applying the power rule, we get:

d/dt (u^(1/3)) = (1/3) * u^(-2/3) * du/dt

Step 2: Determine the derivative of the inner function u = (1 + tan(t)):

The derivative of u with respect to t is:

du/dt = d/dt [ (1 + tan(t)) ] = sec^2(t)

Step 3: Substitute the values back into the chain rule formula:

d/dt (f(t)) = (1/3) * (1 + tan(t))^(-2/3) * sec^2(t)

Therefore, the derivative of f(t) is:

d/dt (f(t)) = (1/3) * (1 + tan(t))^(-2/3) * sec^2(t)