What is the equation of the tangent line of #f(x) =e^x/x-e^(2x-3)/x^3# at #x=2#?
Use the quotient rule to differentiate both functions and then use the difference rule.
We must differentiate the numerator using the chain rule--
As for the denominator:
So, we have
We need to find the y value that the function and the tangent passes through.
Now that we know a point on the function and on the tangent and the slope of the tangent, we can find the equation of the tangent.
Practice exercises:
Hopefully this helps, and good luck!
By signing up, you agree to our Terms of Service and Privacy Policy
To find the equation of the tangent line of the function f(x) = e^x/x - e^(2x-3)/x^3 at x=2, we need to find the slope of the tangent line and a point on the line.
First, let's find the slope. We can do this by finding the derivative of the function f(x).
The derivative of f(x) = e^x/x - e^(2x-3)/x^3 can be found using the quotient rule.
After differentiating and simplifying, we get:
f'(x) = (x^2e^x - 3e^(2x-3) - 2xe^(2x-3))/x^4
Now, let's find the slope at x=2 by substituting x=2 into f'(x):
f'(2) = (2^2e^2 - 3e^(2(2)-3) - 2(2)e^(2(2)-3))/2^4
Simplifying further, we get:
f'(2) = (4e^2 - 3e^1 - 4e^1)/16
f'(2) = (4e^2 - 7e)/16
Now that we have the slope, let's find a point on the line. We can do this by substituting x=2 into the original function f(x):
f(2) = e^2/2 - e^(2(2)-3)/2^3
Simplifying further, we get:
f(2) = e^2/2 - e^1/8
Now we have a point (2, e^2/2 - e^1/8) on the tangent line.
Using the point-slope form of a line, we can write the equation of the tangent line:
y - (e^2/2 - e^1/8) = (4e^2 - 7e)/16 * (x - 2)
Simplifying further, we get the equation of the tangent line:
y = (4e^2 - 7e)/16 * x + (e^2/2 - e^1/8) - (4e^2 - 7e)/8
By signing up, you agree to our Terms of Service and Privacy Policy
To find the equation of the tangent line to ( f(x) = \frac{e^x}{x} - \frac{e^{2x-3}}{x^3} ) at ( x = 2 ), we first need to find the slope of the tangent line, which is the derivative of the function evaluated at ( x = 2 ).
[ f'(x) = \frac{d}{dx} \left( \frac{e^x}{x} - \frac{e^{2x-3}}{x^3} \right) ] [ f'(x) = \frac{e^x \cdot x - e^x - 3e^{2x-3}}{x^4} ]
Now, we can find the slope of the tangent line by evaluating ( f'(x) ) at ( x = 2 ):
[ f'(2) = \frac{e^2 \cdot 2 - e^2 - 3e^1}{2^4} ] [ f'(2) = \frac{2e^2 - e^2 - 3e}{16} ] [ f'(2) = \frac{e^2 - 3e}{16} ]
Now that we have the slope of the tangent line, we can use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is:
[ y - y_1 = m(x - x_1) ]
Substituting ( x_1 = 2 ), ( y_1 = f(2) ), and ( m = f'(2) ):
[ y - f(2) = \frac{e^2 - 3e}{16}(x - 2) ]
We just need to find ( f(2) ) to complete the equation.
[ f(2) = \frac{e^2}{2} - \frac{e^1}{2^3} = \frac{e^2}{2} - \frac{e}{8} ]
Thus, the equation of the tangent line at ( x = 2 ) is:
[ y - \left( \frac{e^2}{2} - \frac{e}{8} \right) = \frac{e^2 - 3e}{16}(x - 2) ]
By signing up, you agree to our Terms of Service and Privacy Policy
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.
- How do you find the equation of the line tangent to #y=3x^2-x^3# at point (1,2)?
- How do you find the average value of the positive y-coordinates of the ellipse #x^2/a^2 + y^2/b^2 = 1#?
- What is the instantaneous velocity of an object moving in accordance to # f(t)= (sqrt(t+2),t+4) # at # t=1 #?
- What is the equation of the tangent line of #f(x)=cosx-e^x/sinx # at #x=pi/3#?
- How do you find the derivative using limits of #f(x)=x^3-12x#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7