4.3 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 two functions f(x) and g(x), then the derivative of their composition is given by: \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).

Front

Find the derivative of \( H(x) = 3x^2 + 2x \) at \( x = 1 \).

Back

\( H'(1) = 9 \) because \( H'(x) = 6x + 2 \) and substituting \( x = 1 \) gives \( H'(1) = 6(1) + 2 = 8 \).

Front

If \( f(x) = (6x^2 - 5x)^7 \), what is \( f'(x) \)?

Back

\( f'(x) = 7(6x^2 - 5x)^6 \cdot (12x - 5) \) using the Chain Rule.

Front

How do you differentiate \( y = \frac{x^3}{x-1} \)?

Back

Using the quotient rule, \( \frac{dy}{dx} = \frac{(3x^2)(x-1) - (x^3)(1)}{(x-1)^2} = \frac{2x^3 - 3x^2}{(x-1)^2} \).

Front

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

Back

Using the Chain Rule, \( f'(x) = 30x^5(x^6 + 4)^4 \).

Front

Find \( y' \) if \( y = \tan(3x^2 + 2) \).

Back

Using the Chain Rule, \( y' = 6x \sec^2(3x^2 + 2) \).

Front

What is the formula for the derivative of a power function \( f(x) = x^n \)?

Back

The derivative is given by \( f'(x) = nx^{n-1} \).

Front

Explain the product rule for differentiation.

Back

The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is: \( (uv)' = u'v + uv' \).

Front

What is the quotient rule for differentiation?

Back

The quotient rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their quotient is: \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \).

Front

How do you find the derivative of a composite function?

Back

Use the Chain Rule: if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).