Calculus math question

Question

If f(x)= 7+5x-4x², find f'(3)

Answer 1

Remember that differentiation is a linear operator, so it applies to each term and each derivative is added together:

d/dx (f(x)) = d/dx (7) + d/dx (5x) - d/dx (4x^2)

What is the derivative of 7, which is just a constant, a lonesome number? Zero.

What is the derivative of 5x, which is a constant times a variable without a power greater than 1? It's just the number part, so 5. (Technically, the power rule is at work here, but the variable x is differentiated as 1*x^(1-1) = 1*x^0 = 1*1 = 1, and you can pull out the 5 to the front before doing differentiation, so 5*1 = 5.)

What is the derivative of 4(x^2), which is a constant times a variable with a power greater than 1? You need the power rule:

d/dx (4(x^2)) = 4 * d/dx (x^2) = 4 * (2x^(2-1)) = 8x

So 0 + 5 - 8x = 8x. So f'(x) = 5 - 8x.

So f'(3) = 5 - 8(3) = 5 - 24 = -19.

I hope this helps!