If f(x) + x^(2)[f(x)]^5 = 34 and f(1) = 2, find f '(1).
To find f '(1), we need to take the derivative of the function f(x) and then evaluate it at x = 1.
Given: f(x) + x^(2)[f(x)]^5 = 34
Step 1: Differentiate both sides of the equation with respect to x using the chain rule.
d/dx [f(x) + x^(2)[f(x)]^5] = d/dx (34)
Step 2: Apply the chain rule to the left side of the equation.
f'(x) + 2x[f(x)]^5 + 5x^2[f(x)]^4 * f'(x) = 0
Step 3: Simplify the equation.
f'(x) + f'(x) * 5x^2[f(x)]^4 + 2x[f(x)]^5 = 0
Step 4: Factor out f'(x).
f'(x) * (1 + 5x^2[f(x)]^4) = -2x[f(x)]^5
Step 5: Solve for f'(x).
f'(x) = -2x[f(x)]^5 / (1 + 5x^2[f(x)]^4)
Step 6: Evaluate f '(1).
Substitute x = 1 into the equation we found in step 5.
f '(1) = -2(1)[f(1)]^5 / (1 + 5(1)^2[f(1)]^4)
Since it is given that f(1) = 2, we can substitute this value into the equation.
f '(1) = -2(1)(2)^5 / (1 + 5(1)^2(2)^4)
Simplifying further:
f '(1) = -64 / (1 + 80)
f '(1) = -64 / 81
Therefore, f '(1) is approximately equal to -0.7901.
To find f '(1), we need to take the derivative of the given equation with respect to x and then evaluate it at x = 1.
Let's start by differentiating both sides of the equation.
We have:
d/dx [f(x) + x^2[f(x)]^5] = d/dx [34]
To differentiate the left side of the equation, we need to apply the product rule twice.
First, let's differentiate f(x) with respect to x:
d/dx [f(x)] = f '(x)
Next, let's differentiate x^2[f(x)]^5 with respect to x:
d/dx [x^2[f(x)]^5] = 2x[f(x)]^5 + x^2 * 5[f(x)]^4 * f '(x)
Using the chain rule, we've multiplied each term by the derivative of f(x) with respect to x, which is f '(x).
Now, let's substitute these derivatives back into the original equation:
f '(x) + (2x[f(x)]^5 + x^2 * 5[f(x)]^4 * f '(x)) = 0
Simplifying this equation, we get:
f '(x) + 2x[f(x)]^5 + 5x^2[f(x)]^4 * f '(x) = 0
Finally, let's evaluate the equation at x = 1, since we need to find f '(1):
f '(1) + 2(1)[f(1)]^5 + 5(1)^2[f(1)]^4 * f '(1) = 0
Substituting f(1) = 2:
f '(1) + 2(1)(2)^5 + 5(1)^2(2)^4 * f '(1) = 0
Simplifying further:
f '(1) + 64 + 40f '(1) = 0
Combining like terms:
41f '(1) + 64 = 0
Now, let's solve for f '(1):
41f '(1) = -64
f '(1) = -64/41
Therefore, f '(1) is equal to -64/41.
f + x^2 f^5 = 34
f' + 2xf^5 + 5x^2 f^4 f' = 0
f' = -2xf^5/(1+5x^2 f^4)
f'(1) = -2(1)(2^5)/(1+5(1)(2^4))
= -64/80
= -4/5