18 flashcards · Shared on 19 August 2026 by AtomAI Library
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What is the derivative of a function at a point?
The instantaneous rate of change of the function at that point
What is the derivative of a function at a point?
The instantaneous rate of change of the function at that point
Using the power rule, what is the derivative of x^7?
7x^6
Which expression is the derivative of 5x^4 - 3x^2 + 8?
20x^3 - 6x
What does the constant multiple rule state?
A constant factor remains as a factor when the function is differentiated
For differentiable functions u(x) and v(x), which is the product rule?
d(uv)/dx = u'v + uv'
What is the derivative of x²e^x?
e^x(x² + 2x)
For differentiable functions u(x) and v(x), where v(x) is nonzero, which formula gives the derivative of u/v?
(u'v - uv')/v²
What is the derivative of (x + 1)/(x - 1)?
-2/(x - 1)²
When is the chain rule used?
When differentiating a composite function
Differentiate (3x + 2)^5 with respect to x.
15(3x + 2)^4
Which expression is the derivative of ln x for x > 0?
1/x
What is the derivative of sin(4x)?
4cos(4x)
What does f'(a) represent geometrically on the graph y = f(x)?
The gradient of the tangent at x = a
A differentiable function has f'(x) > 0 throughout an interval. What does this imply?
The function is increasing on that interval
Suppose f'(a) = 0 and f''(a) > 0. What does the second derivative test indicate?
A local minimum at x = a
How is differentiation commonly used in an optimization problem?
Candidate optimum points are found by solving f'(x) = 0 and checking them appropriately
If s(t) is displacement as a function of time, what do s'(t) and s''(t) represent?
Velocity and acceleration, respectively
A curve has tangent gradient 3 at a point. What is the gradient of the normal there?
-1/3