Find the Normal Line at x=0 y=x^4+9e^x , (0,9)

Math
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Find and evaluate at and to find the slope of the tangent line at and .
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Differentiate both sides of the equation.
The derivative of with respect to is .
Differentiate the right side of the equation.
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Differentiate.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Exponential Rule which states that is where =.
Reform the equation by setting the left side equal to the right side.
Replace with .
Evaluate at and .
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Replace the variable with in the expression.
Simplify each term.
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Raising to any positive power yields .
Multiply by .
Anything raised to is .
Multiply by .
Add and .
The normal line at is perpendicular to the tangent line at . Take the negative reciprocal of the slope of the tangent line to find the slope of the normal line.
Plug in the slope of the tangent line and the and values of the point into the pointslope formula .
Simplify.
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The slope-intercept form is , where is the slope and is the y-intercept.
Rewrite in slope-intercept form.
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Multiply by .
Simplify .
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Subtract from .
Combine and .
Add to both sides of the equation.
Rewrite in slope-intercept form.
Find the Normal Line at x=0 y=x^4+9e^x , (0,9)

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