Exercise 1:
Find the derivative of the function f(x) = x^3
Answer:
Using the power rule, we know that the derivative of x^n is nx^(n-1). So, the derivative of
x^3 is 3x^(3-1) = 3*x^2.
Exercise 2:
Find the derivative of the function f(x) = 2x^2 + 3x - 4
Answer:
Using the sum/difference rule and the power rule, we know that the derivative of 2x^2 is
2*2x^(2-1) = 4x, the derivative of 3x is 3 and the derivative of -4 is 0. So, the derivative of
2x^2 + 3x - 4 is 4x + 3.
Exercise 3:
Find the derivative of the function f(x) = (x^2 + 3x + 2) / (x - 1)
Answer:
Using the quotient rule, we know that the derivative of (x^2 + 3x + 2) / (x - 1) is (x^2 + 3x +
2)' * (x - 1) - (x^2 + 3x + 2) * (x - 1)'. By using the power rule, the derivative of x^2 is 2x and
the derivative of x is 1. So, the numerator becomes (2x+3). And also the derivative of x-1 is
1. So, the denominator becomes 1. Then by simplifying (2x+3)/1 = 2x+3.
Exercise 4:
Find the derivative of the function f(x) = (3x^2 - 4x) * e^x
Answer:
Using the product rule, we know that the derivative of (3x^2 - 4x) * e^x is (3x^2 - 4x)' * e^x +
(3x^2 - 4x) * e^x'. By using the power rule, the derivative of x^2 is 2x and the derivative of x
is 1. So, the first part becomes (6x-4) and the second part is e^x. Then by simplifying (6x-
4)*e^x+e^x(3x^2-4x)=6xe^x+e^x(3x^2-4x).
Exercise 5:
Find the second derivative of the function f(x) = x^3
Answer:
The first derivative of x^3 is 3x^2. Then, by applying the power rule again, the second
derivative is 32x^(2-1) = 6x.
Exercise 6:
Find the third derivative of the function f(x) = x^3
Answer:
The first derivative of x^3 is 3x^2, second derivative is 6x, then by applying the power rule
again, the third derivative is 31x^(1-1) = 3.
