Question Statement

Find the derivative of $y = (3 x - 1)^{4} (- 2 x + 9)^{5}$ with respect to $x$ .

Background and Explanation

This problem involves differentiating a product of two composite functions. You will need to apply the product rule to handle the multiplication, and the chain rule to differentiate the powers of linear expressions.

Solution

Begin by identifying the two functions being multiplied: let $u = (3 x - 1)^{4}$ and $v = (- 2 x + 9)^{5}$ .

Apply the product rule $\frac{d}{d x} (uv) = u^{'} v + u v^{'}$ :

$\frac{d y}{d x} = \frac{d}{d x} (3 x - 1)^{4} \cdot (- 2 x + 9)^{5} + (3 x - 1)^{4} \cdot \frac{d}{d x} (- 2 x + 9)^{5}$

Now apply the chain rule to each derivative. For the first term, differentiate the outer power function and multiply by the derivative of the inner linear function. Do the same for the second term:

$\frac{d y}{d x} = 4 (3 x - 1)^{3} \cdot \frac{d}{d x} (3 x - 1) \cdot (- 2 x + 9)^{5} + (3 x - 1)^{4} \cdot 5 (- 2 x + 9)^{4} \cdot \frac{d}{d x} (- 2 x + 9) = 4 (3 x - 1)^{3} (3 - 0) (- 2 x + 9)^{5} + (3 x - 1)^{4} \cdot 5 (- 2 x + 9)^{4} (- 2 + 0) = 12 (3 x - 1)^{3} (- 2 x + 9)^{5} - 10 (3 x - 1)^{4} (- 2 x + 9)^{4}$

Factor out the greatest common factor $2 (3 x - 1)^{3} (- 2 x + 9)^{4}$ from both terms. Note that the first term retains a factor of $(- 2 x + 9)$ and coefficient $6$ (since $12/2 = 6$ ), while the second term retains a factor of $(3 x - 1)$ and coefficient $- 5$ (since $- 10/2 = - 5$ ):

$\frac{d y}{d x} = 2 (3 x - 1)^{3} (- 2 x + 9)^{4} [6 (- 2 x + 9) - 5 (3 x - 1)] = 2 (3 x - 1)^{3} (- 2 x + 9)^{4} (- 12 x + 54 - 15 x + 5) = 2 (3 x - 1)^{3} (- 2 x + 9)^{4} (- 27 x + 59)$

Key Formulas or Methods Used

Product Rule: $\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$
Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$
Power Rule: $\frac{d}{d x} [x^{n}] = n x^{n - 1}$

Summary of Steps

Identify the two functions in the product: $(3 x - 1)^{4}$ and $(- 2 x + 9)^{5}$
Apply the product rule: $\frac{d}{d x} (uv) = u^{'} v + u v^{'}$
Apply the chain rule to differentiate each composite function, yielding $4 (3 x - 1)^{3} (3)$ and $5 (- 2 x + 9)^{4} (- 2)$
Simplify to obtain $12 (3 x - 1)^{3} (- 2 x + 9)^{5} - 10 (3 x - 1)^{4} (- 2 x + 9)^{4}$
Factor out the common term $2 (3 x - 1)^{3} (- 2 x + 9)^{4}$
Simplify the expression inside the brackets: $6 (- 2 x + 9) - 5 (3 x - 1) = - 27 x + 59$
State the final answer: $2 (3 x - 1)^{3} (- 2 x + 9)^{4} (- 27 x + 59)$

Solution

Begin by identifying the two functions being multiplied: let

u = (3 x - 1)^{4}

and

v = (- 2 x + 9)^{5}

Apply the product rule

\frac{d}{d x} (uv) = u^{'} v + u v^{'}

\frac{d y}{d x} = \frac{d}{d x} (3 x - 1)^{4} \cdot (- 2 x + 9)^{5} + (3 x - 1)^{4} \cdot \frac{d}{d x} (- 2 x + 9)^{5}

Now apply the chain rule to each derivative. For the first term, differentiate the outer power function and multiply by the derivative of the inner linear function. Do the same for the second term:

\frac{d y}{d x} = 4 (3 x - 1)^{3} \cdot \frac{d}{d x} (3 x - 1) \cdot (- 2 x + 9)^{5} + (3 x - 1)^{4} \cdot 5 (- 2 x + 9)^{4} \cdot \frac{d}{d x} (- 2 x + 9) = 4 (3 x - 1)^{3} (3 - 0) (- 2 x + 9)^{5} + (3 x - 1)^{4} \cdot 5 (- 2 x + 9)^{4} (- 2 + 0) = 12 (3 x - 1)^{3} (- 2 x + 9)^{5} - 10 (3 x - 1)^{4} (- 2 x + 9)^{4}

Factor out the greatest common factor

2 (3 x - 1)^{3} (- 2 x + 9)^{4}

from both terms. Note that the first term retains a factor of

(- 2 x + 9)

and coefficient

6

(since

12/2 = 6

), while the second term retains a factor of

(3 x - 1)

and coefficient

- 5

(since

- 10/2 = - 5

\frac{d y}{d x} = 2 (3 x - 1)^{3} (- 2 x + 9)^{4} [6 (- 2 x + 9) - 5 (3 x - 1)] = 2 (3 x - 1)^{3} (- 2 x + 9)^{4} (- 12 x + 54 - 15 x + 5) = 2 (3 x - 1)^{3} (- 2 x + 9)^{4} (- 27 x + 59)

Summary of Steps

Identify the two functions in the product:

(3 x - 1)^{4}

and

(- 2 x + 9)^{5}

Apply the product rule:

\frac{d}{d x} (uv) = u^{'} v + u v^{'}

Apply the chain rule to differentiate each composite function, yielding

4 (3 x - 1)^{3} (3)

and

5 (- 2 x + 9)^{4} (- 2)

Simplify to obtain

12 (3 x - 1)^{3} (- 2 x + 9)^{5} - 10 (3 x - 1)^{4} (- 2 x + 9)^{4}

Factor out the common term

2 (3 x - 1)^{3} (- 2 x + 9)^{4}

Simplify the expression inside the brackets:

6 (- 2 x + 9) - 5 (3 x - 1) = - 27 x + 59

State the final answer:

2 (3 x - 1)^{3} (- 2 x + 9)^{4} (- 27 x + 59)

2.7 Q-3

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.7 Q-3

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps