# 4.1 Implicit Differentiationap Calculus

Problem-Solving Strategy: Implicit Differentiation. To perform implicit differentiation on an equation that defines a function latexy/latex implicitly in terms of a variable latexx/latex, use the following steps: Take the derivative of both sides of the equation. Keep in mind that latexy/latex is a function of latexx/latex. To perform implicit differentiation on an equation that defines a function implicitly in terms of a variable, use the following steps: Take the derivative of both sides of the equation. Keep in mind that is a function of. Consequently, whereas because we must use the Chain Rule to differentiate with respect to. If 5x^2+5x+xy=2 and y(2)= -14 find y'(2) by implicit differentiation. Use implicit differentiation to find an equation of the tangent line to the curve at the given point. X2 + 2xy − y2 + x = 17, (3, 5) (hyperbola) Math. Suppose that x and y are related by the equation x^2/4 + y^3/2 = 4. To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x. Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'. So, when we take the derivative of each term, we get. Alternatively, one can use implicit differentiation a second time to get Substitutmg x = O, y 3, and y' — gives. 1982 Solution (b) = 152 Implicit: 2X—+.

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### 4.1 Implicit Differentiationap Calculus Algebra

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### Section 3-10 : Implicit Differentiation

For problems 1 – 3 do each of the following.

1. Find (y') by solving the equation for y and differentiating directly.
2. Find (y') by implicit differentiation.
3. Check that the derivatives in (a) and (b) are the same.

1. (displaystyle frac{x}{{{y^3}}} = 1) Solution
2. ({x^2} + {y^3} = 4) Solution
3. ({x^2} + {y^2} = 2) Solution

For problems 4 – 9 find (y') by implicit differentiation.

1. (2{y^3} + 4{x^2} - y = {x^6}) Solution
2. (7{y^2} + sin left( {3x} right) = 12 - {y^4}) Solution
3. ({{bf{e}}^x} - sin left( y right) = x) Solution
4. (4{x^2}{y^7} - 2x = {x^5} + 4{y^3}) Solution
5. (cos left( {{x^2} + 2y} right) + x,{{bf{e}}^{{y^{,2}}}} = 1) Solution
6. (tan left( {{x^2}{y^4}} right) = 3x + {y^2}) Solution

### 4.1 Implicit Differentiationap Calculus Multiple Choice

For problems 10 & 11 find the equation of the tangent line at the given point.

1. ({x^4} + {y^2} = 3) at (left( {1, - sqrt 2 } right)). Solution
2. ({y^2}{{bf{e}}^{2x}} = 3y + {x^2}) at (left( {0,3} right)). Solution

### 4.1 Implicit Differentiationap Calculus Calculator

For problems 12 & 13 assume that (x = xleft( t right)), (y = yleft( t right)) and (z = zleft( t right)) and differentiate the given equation with respect to t.

1. ({x^2} - {y^3} + {z^4} = 1) Solution
2. ({x^2}cos left( y right) = sin left( {{y^3} + 4z} right)) Solution