Unit 5 Test Review

Front

What is the formula for the derivative of the function f(x) = (5x^5 + 5)^2?

Back

The derivative is found using the chain rule: f'(x) = 2(5x^5 + 5)(25x^4) = 500x^4(5x^5 + 5).

Front

How do you find the equation of the tangent line to a function at a given point?

Back

To find the equation of the tangent line, calculate the derivative at the given point to find the slope, then use the point-slope form of the line: y - y1 = m(x - x1).

Front

What is the rate of change of the area of a circle with respect to its radius?

Back

The area A of a circle is given by A = πr^2. The rate of change of the area with respect to time is dA/dt = 2πr(dr/dt).

Front

What is the second derivative of an implicit function?

Back

The second derivative of an implicit function can be found by differentiating the first derivative again, applying implicit differentiation rules.

Front

What is the derivative of f(x) = tan^(-1)(4x)?

Back

The derivative is f'(x) = 4/(16x^2 + 1) using the chain rule.

Front

How do you apply the chain rule in differentiation?

Back

The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).

Front

What is the formula for the area of a circle?

Back

The area A of a circle is given by A = πr^2, where r is the radius.

Front

How do you find the slope of a tangent line at a specific point on a curve?

Back

To find the slope of the tangent line, compute the derivative of the function and evaluate it at the specific point.

Front

What is implicit differentiation?

Back

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

Front

What is the relationship between the radius and the area of a circle?

Back

The area of a circle increases with the square of the radius, meaning if the radius increases, the area increases at a rate proportional to the square of the radius.