Differentiation1

Front

What is the derivative of the function f(x) = 1 - x^2?

Back

The derivative is -2x.

Front

Find the derivative of f(x) = \frac{9}{\sqrt[3]{x}}.

Back

The derivative is -3x^{-4/3}.

Front

What is the derivative of the function f(x) = x^3 + x^2 + 3?

Back

The derivative is 3x^2 + 2x.

Front

Calculate the derivative of f(x) = 12 - 8x.

Back

The derivative is -8.

Front

What is the equation of the line normal to f(x) = 5x^5 - 2x at x = -1?

Back

The equation is y + 3 = -\frac{1}{23}(x + 1).

Front

Define differentiation in calculus.

Back

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable.

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = nx^{n-1}.

Front

Explain the product rule in differentiation.

Back

The product rule states that if u(x) and v(x) are functions, then the derivative of their product is u'v + uv'.

Front

What is the quotient rule in differentiation?

Back

The quotient rule states that if u(x) and v(x) are functions, then the derivative of their quotient is (u'v - uv')/v^2.

Front

What does it mean for a function to be differentiable at a point?

Back

A function is differentiable at a point if it has a derivative at that point, meaning it is smooth and has no sharp corners or discontinuities.