Unit Flashcard- Derivatives ( Chain rule)

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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 f(g(x)) is f'(g(x)) * g'(x).

Front

How do you find the derivative of a function at a specific point?

Back

To find the derivative of a function at a specific point, you can use the definition of the derivative, which is the limit of the difference quotient as the interval approaches zero, or apply differentiation rules directly to the function and then substitute the specific point.

Front

What is the derivative of sec(x)?

Back

The derivative of sec(x) is sec(x)tan(x).

Front

How do you differentiate the function v = t + 8/t?

Back

To differentiate v = t + 8/t, apply the power rule and the quotient rule. The derivative is dv/dt = 1 - 8/t^2.

Front

What is the derivative of f(x) = 1/(x^4 + 3x)?

Back

The derivative of f(x) = 1/(x^4 + 3x) is f'(x) = - (x^4 + 3x)^(-2)(4x^3 + 3).

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then the derivative f'(x) = n*x^(n-1), where n is a constant.

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What is 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 (u*v)' = 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 (u/v)' = (u'v - uv')/v^2.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, allowing us to find dy/dx without explicitly solving for y.

Front

What is the derivative of cos(x)?

Back

The derivative of cos(x) is -sin(x).