Questions: Find the equation of the tangent line to f(x)=(x-1)^8 at the point where x=2.

Transcript text: Find the equation of the tangent line to $f(x)=(x-1)^{8}$ at the point where $x=2$. The tangent line equation is

Solution

Step 1: Find the derivative of $f(x)$

The derivative of $f(x)=\left(x - 1\right)^{8}$ with respect to $x$ is $f'(x)=8 \left(x - 1\right)^{7}$.

Step 2: Evaluate the derivative at $x={a_value}$

The slope of the tangent line at $x=2$ is $m = f'(2) = 8$.

Step 3: Find the y-coordinate of the point on the curve

The y-coordinate of the point on the curve at $x=2$ is $y = f(2) = 1$.

Step 4: Use the point-slope form to find the equation of the tangent line

The equation of the tangent line in point-slope form is $y - 1 = 8(x - 2)$, which simplifies to $y = 8 x - 15$.

The equation of the tangent line to the graph of $y=\left(x - 1\right)^{8}$ at $x=2$ is $y = 8 x - 15$.

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