
Find the equation of tangent line to the function y=-1/x^3 at the pointx=1.
Step-by-step solution-:
The given function is:
y=-\frac{1}{x^{3}}
We know that the first derivative of a function at any certain point (x,y) is equal to the slope of the tangent line.
Calculate the value of y for x=1:
y=-\frac{1}{(1)^{3}}
After simplifying we get,
y=-1
Let be m is the slope of the tangent line:
m= First derivative of y at (1,-1):
It means,
m=\frac{dy}{dx}
Find the first derivative of y:
y'=\frac{d(-\frac{1}{x^3})}{dx}=\frac{3}{x^4}
Substitute the value of x=1 into the above expression:
y'=\frac{dy}{dx}=\frac{3}{(1)^4}=3
It means,
m=3
We know that equation of a tangent line is:
(y-y_{1})=m\cdot(x-x_{1})
Substitute the point (1, -1) and slope m=3 into Point-Slope Form:
(y-(-1))=3\cdot(x-1)
After simplifying we get
y+1=3x-3=>y=3x-4
The required equation of the tangent is:
y=3x-4:
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