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What is the equation of the tangent line of #f(x)=(x^2-35)^7# at #x=6#?

Answer 1

Elijah Abram

#y=84x-503#

Find the point the tangent line will intersect.

#f(6)=(6^2-35)^7=1#

The point is #(6,1)#.

Now, to find the slope of the tangent line, find #f'(6)#.

Finding #f'(x)# will require the chain rule:

#f'(x)=7(x^2-35)^6*d/dx[x^2-35]#

#f'(x)=7(x^2-35)^6(2x)#

#f'(x)=14x(x^2-35)^6#

#f'(6)=14(6)(6^2-35)^6=84#

Write the equation of the tangent line in point-slope form:

#y-1=84(x-6)#

In slope-intercept form:

#y=84x-503#

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Answer 2

William Abdalla

To find the equation of the tangent line of f(x)=(x^2-35)^7 at x=6, we need to find the derivative of the function at x=6 and then use the point-slope form of a line.

First, we find the derivative of f(x) using the chain rule:

f'(x) = 7(x^2-35)^6 * 2x

Next, we substitute x=6 into the derivative to find the slope of the tangent line:

f'(6) = 7(6^2-35)^6 * 2(6)

Finally, we can use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope of the tangent line. Plugging in the values, we have:

y - f(6) = f'(6)(x - 6)

Simplifying further will give us the equation of the tangent line.

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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.