Determine the slope of the tangent to the function f(x)=5xe^x at the point with x-coordinate x=2
a) -1/5
b) 10e^2
c) 15e^2
d) -5e^2
what's the trouble? just use the product rule to get
f'(x) = 5e^x + 5xe^x = 5e^x(1+x)
f'(2) = 10e^2
To determine the slope of the tangent to the function f(x)=5xe^x at the point with x-coordinate x=2, we need to find the first derivative of the function and then evaluate it at x=2.
Step 1: Find the first derivative of f(x).
To find the first derivative, we can use the product rule. Let's start by finding the derivative of the first term, 5x, and the derivative of the second term, e^x.
The derivative of 5x is 5.
The derivative of e^x is e^x.
Now we can apply the product rule:
f'(x) = (5)(e^x) + (5x)(e^x)
Simplifying this, we have:
f'(x) = 5e^x + 5xe^x
Step 2: Evaluate f'(x) at x=2.
To find the slope of the tangent at x=2, we substitute x=2 into the expression for f'(x):
f'(2) = 5e^2 + 5(2)e^2
= 5e^2 + 10e^2
= 15e^2
Therefore, the slope of the tangent to the function f(x)=5xe^x at the point with x-coordinate x=2 is 15e^2.
Therefore, the correct answer is:
c) 15e^2
To determine the slope of the tangent to the function f(x) = 5xe^x at the point with x-coordinate x = 2, we need to find the derivative of the function and evaluate it at x = 2.
Step 1: Find the derivative of the function f(x) = 5xe^x.
Using the product rule, the derivative of f(x) is given by:
f'(x) = (5x * e^x)' + (e^x * 5x)'
To find the derivative of 5x, we apply the power rule, which states that the derivative of x^n is n * x^(n-1):
f'(x) = (5 * e^x) + (5x * e^x)
Step 2: Evaluate the derivative at x = 2.
Substitute x = 2 into f'(x):
f'(2) = 5 * e^2 + 5 * 2 * e^2
Step 3: Simplify the expression.
f'(2) = 5e^2 + 10e^2
f'(2) = 15e^2
Therefore, the slope of the tangent to the function f(x) = 5xe^x at the point with x-coordinate x = 2 is 15e^2.
The correct answer is c) 15e^2.