What is the derivative of $$x^4 \, \text{cos} \, x - 4x^3 \, \text{sin} \, x$$ ?

The derivative is $$x^4 \, \text{cos} \, x + 4x^3 \, \text{sin} \, x$$ .

Find $$\frac{dy}{dx}$$ if $$x^3y + xy^3 = -10$$ .

$$\frac{dy}{dx} = -\frac{3x^2y + y^3}{3xy^2 + x^3}$$ .

What is the chain rule for differentiation?

If $$y = f(g(x))$$ , then $$\frac{dy}{dx} = f$$ .

What is the formula for the volume of a sphere?

The volume of a sphere is given by $$V = \frac{4}{3}\pi r^3$$ .

What is the definition of a limit?

A limit is the value that a function approaches as the input approaches a certain value.

What is the Fundamental Theorem of Calculus?

It states that if $$F$$ is an antiderivative of $$f$$ on an interval [a, b], then $$\int_a^b f(x) \, dx = F(b) - F(a)$$ .

What is implicit differentiation?

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x.

What is the quotient rule for differentiation?

If $$u(x)$$ and $$v(x)$$ are functions, then $$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u$$ .

What is the second derivative test for concavity?

If 0"> , the function is concave up; if $$f$$ , the function is concave down.

What is the definition of continuity at a point?

A function is continuous at a point $$c$$ if $$\lim_{x \to c} f(x) = f(c)$$ .

AP Calculus AB Mixed Review Flashcards

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