Find dy/dx cos(6y)=x

Problem

cos(6*y)=x

Solution

1. Differentiate both sides with respect to x using the chain rule for the left side.

d(cos(6*y))/d(x)=d(x)/d(x)

2. Apply the chain rule to the cosine function, noting that y is a function of x

−sin(6*y)⋅(d(6)*y)/d(x)=1

3. Differentiate the inner function 6*y with respect to x

−6*sin(6*y)d(y)/d(x)=1

4. Isolate the derivative d(y)/d(x) by dividing both sides by −6*sin(6*y)

d(y)/d(x)=1/(−6*sin(6*y))

5. Simplify the expression by moving the negative sign to the numerator.

d(y)/d(x)=−1/(6*sin(6*y))

Final Answer

d(y)/d(x)=−1/(6*sin(6*y))

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