c) If [tex]x^3+y^3+6 x y=0[/tex], find [tex]\frac{d y}{d x}[/tex] by implicit differentiation.
2) Given that [tex]8 c x, 7 c x[/tex] and [tex]\frac{2}{6}(6 c x)[/tex] form 3 consecutive terms of an exponential sequence, find:
i) value of [tex]x[/tex]
ii) Common ratio of the sequence
Answer (1)
Use implicit differentiation on x 3 + y 3 + 6 x y = 0 to find d x d y .
Differentiate each term with respect to x , applying the chain rule where necessary.
Solve for d x d y to get d x d y = − 2 x + y 2 x 2 + 2 y .
Analyze the geometric sequence 8 c x , 7 c x , 2 c x and determine that there is no solution for x unless c x = 0 , in which case the common ratio is undefined. d x d y = − 2 x + y 2 x 2 + 2 y
Explanation
Problem Analysis We are given the equation x 3 + y 3 + 6 x y = 0 and asked to find d x d y using implicit differentiation. This means we will differentiate both sides of the equation with respect to x , treating y as a function of x , and then solve for d x d y .
Implicit Differentiation Differentiating both sides of x 3 + y 3 + 6 x y = 0 with respect to x , we get: d x d ( x 3 ) + d x d ( y 3 ) + d x d ( 6 x y ) = d x d ( 0 ) Applying the power rule and chain rule, we have: 3 x 2 + 3 y 2 d x d y + 6 ( x d x d y + y ( 1 ) ) = 0 3 x 2 + 3 y 2 d x d y + 6 x d x d y + 6 y = 0
Solving for dy/dx Now, we isolate the terms containing d x d y :
3 y 2 d x d y + 6 x d x d y = − 3 x 2 − 6 y Factor out d x d y :
d x d y ( 3 y 2 + 6 x ) = − 3 x 2 − 6 y Solve for d x d y :
d x d y = 3 y 2 + 6 x − 3 x 2 − 6 y Simplify by dividing both the numerator and the denominator by 3: d x d y = y 2 + 2 x − ( x 2 + 2 y ) d x d y = − 2 x + y 2 x 2 + 2 y
Geometric Sequence Analysis For the second part of the problem, we are given that 8 c x , 7 c x , and 6 2 ( 6 c x ) are three consecutive terms of a geometric sequence. This simplifies to 8 c x , 7 c x , and 2 c x .
In a geometric sequence, the ratio between consecutive terms is constant. Therefore: 8 c x 7 c x = 7 c x 2 c x Assuming c x = 0 , we can cancel c x from each term: 8 7 = 7 2 This equation is not true, which means there is no value of x that satisfies the condition that 8 c x , 7 c x , and 2 c x form a geometric sequence. However, if c = 0 or x = 0 , then the terms are 0 , 0 , 0 , which can be considered a geometric sequence with an undefined common ratio. Since the problem implies that there is a solution, let's assume c = 0 and x = 0 . Then there is no solution.
Final Answer and Common Ratio Since 8 7 = 7 2 is not true, there is no value for x that makes 8 c x , 7 c x , 2 c x a geometric sequence unless c x = 0 . If c x = 0 , then all terms are 0, and the common ratio is undefined. Assuming c and x are non-zero, there is no solution for x and the common ratio.
Conclusion Therefore, d x d y = − 2 x + y 2 x 2 + 2 y . There is no solution for x that makes 8 c x , 7 c x , 2 c x a geometric sequence unless c x = 0 , in which case the common ratio is undefined.
Examples
Implicit differentiation is used in related rates problems in calculus, such as finding how the volume of a balloon changes with respect to its radius when both are changing with time. Geometric sequences are used in finance to calculate compound interest, where each term represents the amount of money after each compounding period. Understanding these concepts helps in modeling real-world phenomena and making informed decisions.
