,

Differentiate both sides of the equation.

The derivative of with respect to is .

Differentiate the right side of the equation.

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Product Rule which states that is where and .

The derivative of with respect to is .

Simplify the expression.

Move to the left of .

Rewrite as .

Differentiate using the Exponential Rule which states that is where =.

Simplify.

Apply the distributive property.

Multiply by .

Reorder terms.

Reform the equation by setting the left side equal to the right side.

Replace with .

Evaluate at and .

Replace the variable with in the expression.

Simplify each term.

Anything raised to is .

Multiply by .

The exact value of is .

Multiply by .

Anything raised to is .

Multiply by .

The exact value of is .

Multiply by .

Add and .

Plug in the slope of the tangent line and the and values of the point into the point–slope formula .

The slope-intercept form is , where is the slope and is the y-intercept.

Rewrite in slope-intercept form.

Multiply by .

Subtract from .

Add to both sides of the equation.

Find the Tangent Line at (0,4) y=4e^xcos(x) , (0,4)