Find Y\’\’ By Implicit Differentiation Calculator

Calculate y’ and y”

Enter the coefficients, exponents for the equation Ax^m + Byⁿ = C, and a point (x, y) on the curve.

What is a Find y” by Implicit Differentiation Calculator?

A “find y” by implicit differentiation calculator” is a tool used to find the first (y’) and second (y”) derivatives of a function defined implicitly, typically when y cannot be easily expressed as an explicit function of x (like y = f(x)). Our calculator focuses on implicit equations of the form Ax^m + Byⁿ = C, where A, B, C, m, and n are constants.

Implicit differentiation involves differentiating both sides of an equation with respect to x, treating y as a function of x, and then solving for y’ (dy/dx). To find y”, we differentiate y’ again with respect to x, using the rules of differentiation and substituting the expression for y’ where needed.

This calculator is useful for students learning calculus, engineers, and scientists who encounter implicit relations between variables.

Common Misconceptions

• It solves for y explicitly: This calculator finds the derivatives y’ and y” at a specific point (x, y) on the curve, not an explicit function y=f(x).
• It works for any equation: Our calculator is specifically designed for the form Ax^m + Byⁿ = C. More complex implicit equations require more advanced symbolic differentiation.
• The point (x, y) can be anything: The point (x, y) you input must lie on the curve defined by Ax^m + Byⁿ = C for the derivatives to be meaningful at that point.

Find y” by Implicit Differentiation Formula and Mathematical Explanation

For an implicit equation of the form:

Ax^m + Byⁿ = C

We differentiate both sides with respect to x, remembering that y is a function of x (using the chain rule for yⁿ):

A*m*x^m-1 + B*n*y^n-1 * y’ = 0

Solving for y’:

y’ = – (A*m*x^m-1) / (B*n*y^n-1)

To find y”, we differentiate y’ with respect to x using the quotient rule, or as we did in the thought process:

y’ = -(A*m/(B*n)) * x^m-1 * y^1-n

y” = -(A*m/(B*n)) * d/dx(x^m-1 * y^1-n)

y” = -(A*m/(B*n)) * [(m-1)x^m-2y^1-n + x^m-1(1-n)y^-ny’]

Substituting y’:

y” = – (A*m*(m-1)/(B*n)) * x^m-2y^1-n – (A*m/(B*n))² * (n-1) * x^2m-2y^1-2n

y” = – [ A*m*(m-1)*x^m-2*B*n*y^n-1 + A²*m²*(n-1)*x^2m-2*y^n-2 ] / (B*n*y^n-1)² (after some simplification and adjustments)

Let’s use the version: `y” = -(A*m/(B*n)) * (m-1)x^(m-2)y^(1-n) + (A*m/(B*n))^2 * (n-1)x^(2m-2)y^(1-2n)` for calculation as it avoids y” in its own derivation and uses y’ once.

Variables Table

Variable	Meaning	Unit	Typical Range
A	Coefficient of x^m	None (or depends on C)	Any real number
m	Exponent of x	None	Any real number
B	Coefficient of y^n	None (or depends on C)	Any real number (not zero for y’ denominator)
n	Exponent of y	None	Any real number (not zero for y’ denominator)
x	x-coordinate of the point	Depends on context	Any real number
y	y-coordinate of the point	Depends on context	Any real number (not zero if 1-n or 1-2n is negative)
y’	First derivative (dy/dx)	Depends on context	Any real number
y”	Second derivative (d^2y/dx^2)	Depends on context	Any real number

Table of variables used in the find y” by implicit differentiation calculator for Ax^m + Byⁿ = C.

Practical Examples (Real-World Use Cases)

Example 1: Circle x² + y² = 25

Here, A=1, m=2, B=1, n=2, C=25. Let’s find y’ and y” at the point (3, 4).

Inputs: A=1, m=2, B=1, n=2, x=3, y=4.

y’ = – (1*2*3^(2-1)) / (1*2*4^(2-1)) = – (2*3) / (2*4) = -6/8 = -0.75

y” = -(1*2/(1*2)) * (2-1)*3^(2-2)*4^(1-2) + (1*2/(1*2))² * (2-1)*3^(2*2-2)*4^(1-2*2)

y” = -1 * 1 * 3⁰ * 4^-1 + 1² * 1 * 3² * 4^-3

y” = -1 * 1 * 1 * (1/4) + 1 * 9 * (1/64) = -1/4 + 9/64 = -16/64 + 9/64 = -7/64 ≈ -0.109375

Using the alternative y” formula from the circle: y” = -25/y³ = -25/4³ = -25/64 ≈ -0.390625. Let me re-derive y” more carefully for the general case.
`y” = – (A*m*(m-1)/(B*n)) * x^(m-2)y^(1-n) – (A*m/(B*n))^2 * (n-1) * x^(2m-2)y^(1-2n)`
For the circle: `y” = -(1*2*(1)/(1*2))*3^0*4^-1 – (1*2/(1*2))^2 * (1) * 3^2*4^-3 = -1*(1/4) – 1*9*(1/64) = -1/4 – 9/64 = -16/64 – 9/64 = -25/64`. Yes, the second formula `-(n-1)` was wrong, it should be `+(n-1)` then a minus outside.
`y” = -(A*m/(B*n)) * [ (m-1)x^(m-2)y^(1-n) – (n-1)x^(m-1)y^(-n) * (- (A*m*x^(m-1)) / (B*n*y^(n-1))) ]`
`y” = -(A*m*(m-1)/(B*n))x^(m-2)y^(1-n) – (A*m/(B*n))^2 * (n-1)x^(2m-2)y^(-n+1-n) = -(A*m*(m-1)/(B*n))x^(m-2)y^(1-n) – (A*m/(B*n))^2 * (n-1)x^(2m-2)y^(1-2n)`
So for circle: -1*(1/4) – 1*1*9/64 = -1/4 – 9/64 = -25/64. Correct now.

At (3, 4), y’ = -0.75 and y” = -25/64 ≈ -0.390625.

Example 2: Ellipse 4x² + 9y² = 36

Here, A=4, m=2, B=9, n=2, C=36. Let’s find y’ and y” at a point on the ellipse, e.g., x=0, y=2 (0 + 9*4 = 36).

Inputs: A=4, m=2, B=9, n=2, x=0, y=2.

y’ = – (4*2*0^(2-1)) / (9*2*2^(2-1)) = 0 / (18*2) = 0 (horizontal tangent)

y” = -(4*2*(2-1)/(9*2)) * 0^(2-2)*2^(1-2) – (4*2/(9*2))² * (2-1) * 0^(2*2-2)*2^(1-2*2)
If x=0 and m-2=0, we have 0^0 which is tricky. Let’s assume m>2 or m=2, then m-2>=0. If m=2, m-2=0, x^(m-2)=1.
If m=2, 2m-2 = 2, so x^(2m-2)=0. The second term is 0.
y” = -(8/18) * 1 * 0⁰ * 2^-1 = -(4/9) * 1 * (1/2) = -2/9 (assuming 0^0=1 when m=2).
If m>2, then m-2>0, x^(m-2)=0, 2m-2>0, x^(2m-2)=0, so y”=0.
For m=2: y” = -(4/9) * 1 * (1/2) – 0 = -2/9 ≈ -0.222

How to Use This Find y” by Implicit Differentiation Calculator

1. Enter Coefficients and Exponents: Input the values for A, m, B, and n corresponding to your equation Ax^m + Byⁿ = C.
2. Enter Point Coordinates: Input the x and y coordinates of the point on the curve where you want to find y’ and y”. Make sure the point satisfies the equation.
3. Calculate: Click the “Calculate” button. The calculator will compute y’ and y”.
4. View Results: The calculator will display y’ and y” (as the primary result), along with some intermediate values used in the calculation. A bar chart will also show y’ and y”.
5. Reset: Click “Reset” to clear the fields to their default values for a new calculation with our find y” by implicit differentiation calculator.
6. Copy Results: Click “Copy Results” to copy the main outputs and inputs to your clipboard.

Understanding the results helps in analyzing the slope (y’) and concavity (y”) of the curve at the given point.

Key Factors That Affect y” Results

• Coefficients A and B: These scale the influence of the x and y terms, affecting both y’ and y”.
• Exponents m and n: These determine the power of x and y and significantly influence the derivatives, especially their complexity.
• Point (x, y): The values of x and y at which the derivatives are evaluated are crucial, as y’ and y” are functions of x and y.
• Denominator B*n*y^n-1: If this term is close to zero, y’ becomes very large (vertical tangent), and y” will also be affected. Division by zero indicates an undefined y’ at that point or along that y-value if n>1.
• Values of m-1, m-2, n-1, 1-n, 1-2n: The signs and magnitudes of these exponents determine how x and y influence the derivatives, especially near x=0 or y=0 if these exponents are negative.
• Correctness of the Point: If the point (x, y) does not actually lie on the curve Ax^m + Byⁿ = C, the calculated y’ and y” are mathematically valid for the formulas but don’t represent the derivatives *of the curve* at that point.

Frequently Asked Questions (FAQ)

What if my equation is not in the form Ax^m + Byⁿ = C?

This specific find y” by implicit differentiation calculator is designed only for equations of the form Ax^m + Byⁿ = C. For other forms, you would need a more general symbolic differentiator or perform the steps manually.

What if B*n*y^n-1 = 0?

If B*n*y^n-1 = 0 (and A*m*x^m-1 is not zero), y’ is undefined, suggesting a vertical tangent line to the curve at that point. Our calculator might show Infinity or an error.

What if y=0 and some exponents of y are negative?

If exponents like 1-n or 1-2n are negative, y cannot be zero, as it would lead to division by zero in the y’ or y” formulas. Ensure y is non-zero if n>1.

Can I find y”’ using this method?

Yes, you could differentiate the expression for y” with respect to x, substituting y’ and y” where they appear, to find y”’. However, this calculator does not compute y”’.

How do I know if the point (x, y) is on the curve?

Substitute your x and y values into the equation Ax^m + Byⁿ. If the result is close to C (allowing for minor rounding), the point is on or near the curve.

Why is y” important?

The second derivative, y”, tells us about the concavity of the curve. If y” > 0, the curve is concave up; if y” < 0, it's concave down. It's also used in optimization problems (second derivative test) and physics (acceleration).

Can I use this find y” by implicit differentiation calculator for explicit functions y=f(x)?

Yes, if you can rewrite y=f(x) as f(x) – y = 0 or something similar fitting Ax^m + Byⁿ = C (though it might be trivial, like 1*y^1 = C where C is f(x), which doesn’t fit the form). It’s easier to differentiate y=f(x) directly twice.

What if m or n are fractions?

The formulas still apply, but you need to be careful with the domains of x and y (e.g., x must be non-negative if m-2 is fractional like 1/2).

Related Tools and Internal Resources

• Derivative Calculator: Find the derivative of explicit functions.
• Integral Calculator: Calculate definite and indefinite integrals.
• Implicit Differentiation Explained: A guide to understanding the process of implicit differentiation.
• Second Derivative Test: Learn how y” is used in finding local maxima and minima.
• Equation Solver: Solve various types of equations.
• Function Grapher: Plot graphs of functions.