Find dy/dx tan(4x+y)=4x

Problem

tan(4*x+y)=4*x

Solution

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

d(tan(4*x+y))/d(x)=(d(4)*x)/d(x)

2. Apply the chain rule to the tangent function, noting that the derivative of tan(u) is sec2(u)

sec2(4*x+y)⋅d(4*x+y)/d(x)=4

3. Differentiate the inner function with respect to x treating y as a function of x

sec2(4*x+y)⋅(4+d(y)/d(x))=4

4. Distribute the sec2(4*x+y) term to isolate the derivative.

4*sec2(4*x+y)+sec2(4*x+y)d(y)/d(x)=4

5. Isolate the term containing d(y)/d(x) by subtracting 4*sec2(4*x+y) from both sides.

sec2(4*x+y)d(y)/d(x)=4−4*sec2(4*x+y)

6. Solve for dy/dx by dividing both sides by sec2(4*x+y)

d(y)/d(x)=(4−4*sec2(4*x+y))/sec2(4*x+y)

7. Simplify the expression by splitting the fraction and using the identity cos2(u)=1/sec2(u)

d(y)/d(x)=4/sec2(4*x+y)−(4*sec2(4*x+y))/sec2(4*x+y)

d(y)/d(x)=4*cos2(4*x+y)−4

8. Factor out the constant and apply the Pythagorean identity cos2(u)−1=−sin2(u)

d(y)/d(x)=4*(cos2(4*x+y)−1)

d(y)/d(x)=−4*sin2(4*x+y)

Final Answer

d(y)/d(x)=−4*sin2(4*x+y)

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