1.) solve for x, 2 = (2x+2)e^x
2.) If you forget the expression that results when differentiating the tangent function. Explain how you can derive this derivative using another method?
Note: question 2 has no correlation with question 1, also no mistakes were made with question 2.
(2x+2)e^x = 2
(x+1) e^x = 1
by inspection, x=0
tanx = sinx/cosx
so use the quotient rule
1.) To solve for x in the equation 2 = (2x + 2)e^x, we can follow these steps:
Step 1: Expand the equation by distributing the e^x to the terms in the parentheses:
2 = 2x e^x + 2e^x
Step 2: Move all terms to one side of the equation:
2x e^x + 2e^x - 2 = 0
Step 3: Factor out the common factor of e^x:
e^x(2x + 2) - 2 = 0
Step 4: Simplify the equation further:
2x e^x + 2 e^x - 2 = 0
Step 5: Divide both sides of the equation by e^x:
2x + 2 - 2/e^x = 0
Step 6: Combine like terms:
2x - 2/e^x = -2
Step 7: Add 2/e^x to both sides of the equation:
2x = 2/e^x - 2
Step 8: Simplify further:
2x = 2 (1/e^x - 1)
Step 9: Divide both sides by 2:
x = 1/e^x - 1
Now, to solve for x, we can use numerical methods or approximation techniques like Newton's method or the bisection method since the equation is not easily solvable algebraically.
2.) If you forget the expression that results when differentiating the tangent function, you can derive its derivative using another method called the quotient rule.
The tangent function can be defined as tan(x) = sin(x)/cos(x). To find its derivative, we can apply the quotient rule:
Step 1: Differentiate the numerator and denominator separately.
The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x).
Step 2: Apply the quotient rule, which states that the derivative of the quotient of two functions is given by (denominator * derivative of the numerator - numerator * derivative of the denominator) divided by (denominator)^2.
Applying the quotient rule to tan(x) = sin(x)/cos(x), we get:
[cos(x) * cos(x) - sin(x) * (-sin(x))] / cos^2(x)
Simplifying this expression, we obtain:
cos^2(x) + sin^2(x) / cos^2(x)
Using the trigonometric identity cos^2(x) + sin^2(x) = 1, the expression becomes:
1 / cos^2(x)
Recalling that cos^2(x) can also be written as sec^2(x), the final result is:
1 / cos^2(x) = sec^2(x)
Therefore, the derivative of the tangent function is sec^2(x).