Question

in exercises 3–24, use the rules of differentiation to find the derivative of the function.

1. ( y = 12 ) 4. ( f(x) = -9 )
2. ( y = x^7 ) 6. ( y = x^{16} )
3. ( y = \frac{1}{x^5} ) 8. ( y = \frac{1}{x^8} )
4. ( f(x) = sqrt5{x} ) 10. ( g(x) = sqrt4{x} )
5. ( f(x) = x + 11 ) 12. ( g(x) = 3x - 1 )
6. ( f(t) = -2t^2 + 3t - 6 ) 14. ( y = t^2 + 2t - 3 )
7. ( g(x) = x^2 + 4x^3 ) 16. ( y = 8 - x^3 )
8. ( s(t) = t^3 + 5t^2 - 3t + 8 ) 18. ( f(x) = 2x^3 - x^2 + 3x )

Explanation:

To solve these derivative problems, we'll use basic differentiation rules like the constant rule, power rule, and sum/difference rule. Let's solve a few as examples (e.g., problem 3, 5, 7, 11):

Problem 3: \( y = 12 \)

Step1: Apply Constant Rule

The derivative of a constant \( c \) is \( 0 \), so \( \frac{d}{dx}(12) = 0 \).

Step1: Apply Power Rule

The power rule states \( \frac{d}{dx}(x^n) = nx^{n-1} \). For \( n = 7 \), we get:
\( \frac{d}{dx}(x^7) = 7x^{7-1} = 7x^6 \).

Step1: Rewrite as Power

Rewrite \( \frac{1}{x^5} \) as \( x^{-5} \).

Step2: Apply Power Rule

Using \( \frac{d}{dx}(x^n) = nx^{n-1} \) with \( n = -5 \):
\( \frac{d}{dx}(x^{-5}) = -5x^{-5-1} = -5x^{-6} = -\frac{5}{x^6} \).

Answer:

\( 0 \)

Problem 5: \( y = x^7 \)