Let y=f(x) be a twice differentiable function such that f(1)=3 and dy/dx= (4y^2+7x^2)^1/2. What is the value of d2y/dx2 at x=1?

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Algebra I

Point-Slope Form: y - y₁ = m(x - x₁)  

x₁ - x coordinate y₁ - y coordinate m - slope

Calculus

Derivatives

Derivative Notation

Taking Derivatives with respect to x

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Chain Rule:

Step-by-step explanation:

Step 1: Define

Function: y = f(x), twice differentiable.

Step 2: Differentiate

Remember we are taking the derivative with respect to x.

Chain Rule [Basic Power Rule]:                                                                     [2nd Derivative] Simplify:                                                                               [2nd Derivative - Chain Rule] Basic Power Rule:                                         [2nd Derivative] Simplify:                                                                               [2nd Derivative] Simplify:                                                                              

Step 3: Evaluate

We are given x = 1 and f(1) = 3. This will tell us the instantaneous slope.

Substitute [2nd Deriv]:                     [√Radical] Exponents:                     [Fraction] Multiply:                           [√Radical] Add:                              

This tells us that the rate of change of the slope of the tangent line is .

We can also write an equation for the instantaneous slope:

[Equation]