Question

If 62+7f(x)+ 6x² (f(x))³ = 0 and f(1)=−2, find f′(1). f′(1)=

Answer 1

To determine f'(1), take the derivative of both sides of the equation with respect to x. Then substitute x=1 and f(1)=-2 into the derived formula, and solve for f'(1) we get f(1)=-2.

Step-by-step explanation:

To find f'(1), where f(x) is a function defined implicitly by an equation, we need to take the derivative of the entire equation with respect to x. The given equation is 62 + 7f(x) + 6x²(f(x))³ = 0. Using the Chain Rule and Power Rule for differentiation, we can find f'(x) in terms of x and f(x), and then substitute x=1 and f(1)=-2 to get f'(1).

Here's the step-by-step differentiation:

1. Derivative of the constant 62 is 0.
2. Derivative of 7f(x) with respect to x is 7f'(x) because of the Chain Rule.
3. For the third term 6x²(f(x))³, we apply the Product Rule: the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
4. We find the derivative of 6x² which is 12x, and then multiply it by (f(x))³. We also multiply 6x² by the derivative of (f(x))³, which is 3(f(x))²f'(x) using the Chain Rule.
5. After taking the derivative, we get 0 = 7f'(x) + 12x(f(x))³ + 18x²(f(x))²f'(x).
6. Solving for f'(x), we plug in the known values x=1 and f(1)=-2.

Aligning with the initial conditions, f'(1) will be found from solving the resulting equation for f'(x) after substitution.