Final answer:
To find the tangent line to the curve at x=1 for the function f(x)=e^{x^3-1}, calculate the derivative, evaluate it at x=1 to get the slope, and then use the point-slope form. The equation of the tangent line at x=1 is y = 3x - 2.
Step-by-step explanation:
To find the equation of the tangent line to the curve y=f(x) at a specific value of x, we need to compute the derivative of the function to find the slope of the tangent at that point. For the function f(x) = e^{x^3-1}, we find the derivative, evaluate it at x = 1, and then use the point-slope form of the line to write the equation of the tangent.
Let's calculate the derivative:
f'(x) = d/dx (e^{x^3-1})
= e^{x^3-1} × d/dx (x^3 - 1) by the chain rule
= e^{x^3-1} × 3x^2
Now, evaluating the derivative at x = 1:
f'(1) = e^{1^3-1} × 3 × 1^2
= e^0 × 3
= 3
The slope of the tangent line at x = 1 is 3, and since we are given the x-value, the point on the curve is (1, f(1)).
f(1) = e^{1^3 - 1}
= e^0
= 1
So the point is (1, 1). Using the point-slope form, y - y1 = m(x - x1), we get:
y - 1 = 3(x - 1)
Thus, the equation of the tangent line at x = 1 is y = 3x - 2.