Compute f '(a) algebraically for the given value of a
5x^2+x; a = -9
f'(x) = 10x+1 for any x. so,
f'(-9) = -89
To compute f'(a) algebraically, we need to find the derivative of the function f(x) = 5x^2 + x.
Step 1: Start with the function f(x) = 5x^2 + x
Step 2: Apply the power rule for differentiation. For any term of the form ax^n, the derivative is given by d/dx(ax^n) = anx^(n-1).
Step 3: Differentiate each term of the function separately.
- Differentiate the first term, 5x^2:
d/dx(5x^2) = 5 * 2x^(2-1) = 10x
- Differentiate the second term, x:
d/dx(x) = 1
Step 4: Combine the derivatives of each term:
f'(x) = 10x + 1
Step 5: Substitute the given value a = -9 into the derivative:
f'(-9) = 10(-9) + 1 = -90 + 1 = -89
Therefore, f'(-9) = -89 when a = -9.
To compute f '(a) algebraically, we need to find the derivative of the function f(x) = 5x^2 + x, and then evaluate it at the given value a = -9.
Step 1: Find the derivative of f(x):
To differentiate 5x^2, we use the Power Rule for differentiation. According to the Power Rule, the derivative of x^n, where n is any real number, is nx^(n-1).
So, the derivative of 5x^2 is 10x^1, which can be simplified to 10x.
To differentiate x, we use the Power Rule again. The derivative of x^1 is 1x^(1-1) = 1.
Therefore, the derivative of f(x) = 5x^2 + x is f'(x) = 10x + 1.
Step 2: Evaluate f '(a) at a = -9:
To find f '(a), substitute a = -9 into the derivative function f'(x) = 10x + 1.
f '(a) = 10(-9) + 1
= -90 + 1
= -89
So, f '(-9) = -89.