How do you find the equation of the tangent line to graph #y=(lnx)^cosx# at the point #(e, 1)#?

Answer 1

The equation is #y = cose/ex - cose +1#

#lny = ln(lnx)^cosx#
#lny = cosxln(lnx)#
Differentiate #ln(lnx)# using the chain rule.
Let #y = lnu# and #u = lnx#.
#dy/(du) = 1/u# and #(du)/dx = 1/x#
#dy/dx = dy/(du) xx (du)/dx#
#dy/dx= 1/u xx 1/x#
#dy/dx = 1/lnx xx 1/x#
#dy/dx = 1/(xlnx)#

We can differentiate the entire function using the product rule.

#1/y(dy/dx) = -sinxln(lnx) + cosx/(xlnx)#
#dy/dx = y(-sinxln(lnx) + cosx/(xlnx))#
#dy/dx= (lnx)^cosx(-sinxln(lnx) + cosx/(xlnx))#
Now, determine the slope of the tangent by inserting your point #x= a# within the derivative.
#m_"tangent" = (lne)^cos(e)(-sin(e)ln(lne) + cose/(elne))#
#m_"tangent" = 1(cos(e))/e#
#m_"tangent" = cose/e#

We now determine the equation of the tangent line.

#y - y_1 = m(x- x_1)#
#y - 1 = cose/e(x - e)#
#y = cose/ex - cose +1#

Hopefully this helps!

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Answer 2

To find the equation of the tangent line to the graph of y = (lnx)^cosx at the point (e, 1), we need to find the slope of the tangent line and then use the point-slope form of a linear equation.

  1. Find the derivative of the function y = (lnx)^cosx using the chain rule: dy/dx = (cosx * (lnx)^(cosx-1) * (1/x)) - ((lnx)^cosx * sinx)

  2. Evaluate the derivative at x = e: dy/dx = (cos(e) * (ln(e))^(cos(e)-1) * (1/e)) - ((ln(e))^cos(e) * sin(e))

  3. Simplify the expression: dy/dx = (cos(e) * (1/e)) - ((ln(e))^cos(e) * sin(e))

  4. Substitute the x-coordinate of the given point (e, 1) into the derivative to find the slope: m = (cos(e) * (1/e)) - ((ln(e))^cos(e) * sin(e))

  5. Use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point (e, 1): y - 1 = m(x - e)

  6. Simplify the equation: y - 1 = [(cos(e) * (1/e)) - ((ln(e))^cos(e) * sin(e))](x - e)

This is the equation of the tangent line to the graph of y = (lnx)^cosx at the point (e, 1).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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