Chain Rule Practice

Front

What is the Chain Rule in calculus?

Back

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function \( f(g(x)) \), then the derivative is given by \( f'(g(x))g'(x) \).

Front

When should you use the Chain Rule?

Back

You should use the Chain Rule when differentiating a function that is composed of another function, i.e., when there is a function within a function.

Front

Find \( \frac{dp}{dq} \) if \( p = \frac{1}{\sqrt{q+9}} \).

Back

\( -\frac{1}{2(q+9)^{\frac{3}{2}}} \)

Front

What is the derivative of \( f(x) = (x^6 + 4)^5 \)?

Back

\( f'(x) = 30x^5(x^6 + 4)^4 \)

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, you need the slope of the function at that point (found using the derivative) and the point itself. The equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope.

Front

What does \( f'(g(x))g'(x) \) represent?

Back

It represents the derivative of the composition of two functions, where \( f \) is the outer function and \( g \) is the inner function.

Front

What is the derivative of \( f(x) = (5x^5 + 5)^2 \)?

Back

\( f'(x) = 500x^4(5x^5 + 5) \)

Front

Explain the concept of implicit differentiation.

Back

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to a variable.

Front

What is the significance of the derivative in calculus?

Back

The derivative represents the rate of change of a function with respect to a variable. It provides information about the slope of the tangent line to the graph of the function at any given point.

Front

How do you apply the Chain Rule to find the derivative of a composite function?

Back

To apply the Chain Rule, identify the outer and inner functions, differentiate the outer function while keeping the inner function unchanged, and then multiply by the derivative of the inner function.