Derivatives and Differentiation Rules
Applying basic rules to find rates of change
Derivatives and Differentiation Rules
Applying basic rules to find rates of change
Math - Grade 9-12
- 1
Find the derivative of f(x) = 7.
A horizontal line has no change in y as x changes.
The derivative is f'(x) = 0 because the derivative of a constant is always 0. - 2
Find the derivative of f(x) = x^5.
The derivative is f'(x) = 5x^4 by the power rule. - 3
Find the derivative of f(x) = 4x^3 - 2x + 9.
Differentiate each term separately.
The derivative is f'(x) = 12x^2 - 2 because the derivative of 4x^3 is 12x^2, the derivative of -2x is -2, and the derivative of 9 is 0. - 4
Find the derivative of f(x) = 6x^4 + x^2 - 8x + 1.
The derivative is f'(x) = 24x^3 + 2x - 8. - 5
Find the derivative of f(x) = 3x^2(x + 5).
Use the product rule: (uv)' = u'v + uv'.
The derivative is f'(x) = 6x(x + 5) + 3x^2(1), which simplifies to 9x^2 + 30x. This uses the product rule. - 6
Find the derivative of f(x) = (x^2 + 1)(x^3 - 4).
The derivative is f'(x) = 2x(x^3 - 4) + (x^2 + 1)(3x^2). This simplifies to 5x^4 + 3x^2 - 8x. - 7
Find the derivative of f(x) = (2x + 1)/(x - 3).
Use the quotient rule: (u/v)' = (u'v - uv')/v^2.
The derivative is f'(x) = ((2)(x - 3) - (2x + 1)(1)) / (x - 3)^2. This simplifies to -7/(x - 3)^2. - 8
Find the derivative of f(x) = (x^2 + 4)/x.
The derivative is f'(x) = (2x(x) - (x^2 + 4)(1)) / x^2, which simplifies to (x^2 - 4)/x^2. An equivalent form is 1 - 4/x^2. - 9
Find the derivative of f(x) = (3x - 2)^4.
Differentiate the outside function first, then multiply by the derivative of the inside.
The derivative is f'(x) = 4(3x - 2)^3(3), which simplifies to 12(3x - 2)^3. This uses the chain rule. - 10
Find the derivative of f(x) = (x^2 + 5x)^3.
The derivative is f'(x) = 3(x^2 + 5x)^2(2x + 5). - 11
Find the derivative of f(x) = sqrt(x), written as x^(1/2).
Rewrite the square root using an exponent and apply the power rule.
The derivative is f'(x) = (1/2)x^(-1/2), which can also be written as 1/(2sqrt(x)). - 12
Find the derivative of f(x) = 1/x^3, written as x^(-3).
The derivative is f'(x) = -3x^(-4), which can also be written as -3/x^4. - 13
A particle moves so that its position is s(t) = 2t^3 - 9t^2 + 12t. Find the velocity function v(t).
Differentiate the position function term by term.
The velocity function is v(t) = s'(t) = 6t^2 - 18t + 12 because velocity is the derivative of position. - 14
The cost function is C(x) = 5x^2 + 10x + 200. Find C'(x) and explain what it represents.
The derivative is C'(x) = 10x + 10. It represents the rate at which cost changes with respect to the number of items produced, also called the marginal cost. - 15
Find the derivative of f(x) = x^3 - 4x^2 + 7x - 6, then evaluate f'(2).
Differentiate first, then substitute x = 2.
First, f'(x) = 3x^2 - 8x + 7. Then f'(2) = 3(4) - 8(2) + 7 = 12 - 16 + 7 = 3.