Answer :
If f''(x) is f(x) =x⁷-x⁵+e³-x+eˣ, the correct equation representing f''(x) is most likely: 7x⁶-5x⁴+3e³-x+eˣ. The correct option is d.
Let's delve deeper into finding the second derivative (f''(x)) of the given function (f(x)) = x⁷-x⁵+e³-x+eˣ. We'll employ differentiation to arrive at the answer.
Step-by-Step Differentiation:
First Derivative (f'(x)):
We begin by finding the first derivative of f(x). Remember that the derivative represents the rate of change of the function. Here's how we differentiate each term:
d(x⁷)/dx = 7x⁶ (using the power rule: d(x^n)/dx = nx^(n-1))
d(-x⁵)/dx = -5x⁴ (power rule)
d(e³)/dx = 0 (since e is a constant and its derivative is zero)
d(-x)/dx = -1 (derivative of a constant is zero)
d(eˣ)/dx = eˣ (exponential rule)
Summing the derivatives of all terms, we obtain the first derivative:
f'(x) = 7x⁶ - 5x⁴ - 1 + eˣ
Second Derivative (f''(x)):
Now, we differentiate f'(x) to find the second derivative. The second derivative represents the rate of change of the rate of change, essentially describing how the function's slope is changing. Here's the differentiation process for each term in f'(x):
d(7x⁶)/dx = 42x⁵ (power rule)
d(-5x⁴)/dx = -20x³ (power rule)
d(-1)/dx = 0 (derivative of a constant is zero)
d(eˣ)/dx = eˣ (exponential rule)
Adding the derivatives of all terms, we get the second derivative:
f''(x) = 42x⁵ - 20x³ + eˣ
Verifying Answer Choices:
By comparing the obtained second derivative (f''(x)) with the answer choices:
Option a) misses the constant term (-1) from the first derivative.
Option b) includes an extra term (3e³) that doesn't appear in the differentiation process.
Option c) misses the constant term (-1) and has an extra term (3e²).
Option d) perfectly matches our derived second derivative: f''(x) = 42x⁵ - 20x³ + eˣ.
Therefore, option d) is the correct answer.
