Derivatives of Polynomials

Front

What is the derivative of a polynomial function?

Back

The derivative of a polynomial function is obtained by applying the power rule, which states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).

Front

Find \( h'(x) \) if \( h(x) = 5x^3 - 2x \).

Back

\( h'(x) = 15x^2 - 2 \)

Front

What is the derivative of \( x^n \)?

Back

The derivative of \( x^n \) is \( nx^{n-1} \).

Front

Differentiate \( h(x) = 3x^{5/3} - 6x^{1/6} + 3x \).

Back

\( h'(x) = 5x^{2/3} - x^{-5/6} + 3 \)

Front

Let \( f(x) = 1 - 3x^2 \). What is \( f'(1) \)?

Back

\( f'(1) = -6 \)

Front

How do you arrange the polynomial \( 6x^2 + 7x^4 + 5x^3 + 15 - 8x \) in descending order?

Back

The correct order is \( 7x^4 + 5x^3 + 6x^2 - 8x + 15 \).

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

Differentiate \( f(x) = 4x^5 - 2x^3 + x \).

Back

\( f'(x) = 20x^4 - 6x^2 + 1 \)

Front

What is the derivative of a constant?

Back

The derivative of a constant is 0.

Front

If \( f(x) = 2x^4 + 3x^2 - 5 \), what is \( f'(x) \)?

Back

\( f'(x) = 8x^3 + 6x \)