Product & Chain Rule

Front

Define the Product Rule in calculus.

Back

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

Front

Define the Chain Rule in calculus.

Back

The Chain Rule is used to differentiate composite functions. If \( y = f(g(x)) \), then the derivative is given by: \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

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Back

Using the Product Rule, the derivative is: \( y' = (2)(5x - 4) + (2x + 1)(5) = 20x - 8 + 10x + 5 = 20x - 3 \).

Front

Back

Using the Product Rule, the derivative is: \( y' = 2x^2(3) + (2)(2x)(3x - 4) = 6x^2 - 16x \).

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Back

Using the Product Rule, the derivative is: \( y' = (2x)(x + 7) + (x^2 - 1)(1) = 2x^2 + 14x + x^2 - 1 = 3x^2 + 14x - 1 \).

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Back

Using the Chain Rule, the derivative is: \( y' = cos(2x^2) \cdot (4x) = 4x cos(2x^2) \).

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Back

Using the Product Rule, the derivative is: \( y' = (3x^4 - 7)(10x) + (12x^3)(5x^2 + 1) \).

Front

What is the derivative of a constant multiplied by a function?

Back

If \( c \) is a constant and \( f(x) \) is a function, then \( \frac{d}{dx}[cf(x)] = c \cdot f'(x) \).

Front

What is the derivative of a sum of functions?

Back

If \( f(x) \) and \( g(x) \) are functions, then \( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \).

Front

Back

The derivative is: \( y' = 3x^2 + 10x - 2 \).