Find Where dy/dx is Equal to Zero x=sec(1/y)

Problem

x=sec(1/y)

Solution

1. Differentiate implicitly with respect to x by applying the chain rule to the right side of the equation.

d(x)/d(x)=d(sec(1/y))/d(x)

2. Apply the derivative of the secant function, which is sec(u)*tan(u) and multiply by the derivative of the inner function 1/y

1=sec(1/y)*tan(1/y)d()/d(x)1/y

3. Differentiate the reciprocal 1/y using the power rule and the chain rule, resulting in −1/(y2)d(y)/d(x)

1=sec(1/y)*tan(1/y)*(−1/(y2)d(y)/d(x))

4. Isolate the derivative d(y)/d(x) by multiplying both sides by −y2 and dividing by the trigonometric terms.

d(y)/d(x)=(−y2)/(sec(1/y)*tan(1/y))

5. Analyze the condition for the derivative to be zero. A fraction is zero only when its numerator is zero and its denominator is non-zero.

−y2=0

6. Solve for y to find the value that makes the numerator zero.

y=0

7. Check for validity by substituting y=0 back into the original equation x=sec(1/y) Since division by zero is undefined, the expression 1/0 does not exist.

8. Conclude that there are no real values of y (and thus no values of x where the derivative is equal to zero because the only candidate for the numerator being zero makes the original function undefined.

Final Answer

d(y)/d(x)=0⇒No solution

Want more problems? Check here!