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In this lesson, we will learn how to find the derivative of a function using the product rule.

Q1:

Find the first derivative of the function π¦ = π₯ ( 4 π₯ + 9 ) 4 9 at π₯ = β 2 .

Q2:

Find the first derivative of the function π¦ = 2 π₯ ( 8 π₯ + 7 ) 5 3 at π₯ = β 1 .

Q3:

Find the first derivative of the function π¦ = 3 π₯ ( 9 π₯ + 8 ) 5 9 at π₯ = β 1 .

Q4:

Find the first derivative of the function π¦ = 2 π₯ π 8 5 π₯ .

Q5:

Given that d d π₯ π = π π π π₯ π π₯ , find π so that π ( π₯ ) = π₯ π 2 π π₯ satisfies π β² ( 1 ) = 0 .

Q6:

Find the first derivative of the function π ( π₯ ) = οΉ 2 π₯ + π₯ β 5 ο οΌ π₯ + 3 β π₯ β 3 π₯ ο 4 2 .

Q7:

Let π ( π₯ ) = β 3 π ( π₯ ) [ β ( π₯ ) β 1 ] . If π β² ( β 4 ) = β 1 , β β² ( β 4 ) = β 9 , β ( β 4 ) = β 6 , and π ( β 4 ) = β 1 , find π β² ( β 4 ) .

Q8:

Find d d π¦ π₯ at π₯ = 2 when π¦ = οΉ π₯ + π₯ β 2 ο οΉ β 3 π₯ + 7 π₯ β 1 ο 2 2 2 5 .

Q9:

Find d d π¦ π₯ if π¦ = π π 2 π₯ 3 π₯ + 8 2 .

Q10:

Suppose that π is differentiable. What is the derivative of π₯ π ( π₯ ) 3 ?

Q11:

The product rule says that ( π π ) = π π + π π β² β² β² . Use this to derive a formula for the derivative ( π π β ) β² .

Q12:

Using the product rule, find d d π₯ ( π₯ π ) 2 β π₯ .

Q13:

Suppose that π ( 2 ) = 3 , π ( 2 ) = 5 , π β² ( 2 ) = β 1 , and π β² ( 2 ) = 6 . Evaluate ( π ( π₯ ) π ( π₯ ) ) β² β π β² ( π₯ ) π β² ( π₯ ) at π₯ = 2 .

Q14:

Find the first derivative of π¦ = ( π₯ β 5 ) ( π₯ β 2 ) 6 at ( 1 , β 4 ) .

Q15:

Find the first derivative of π ( π₯ ) = οΉ 9 π₯ β π₯ β 7 ο οΉ 7 π₯ β 8 π₯ β 7 ο 2 2 at ( β 1 , 2 4 ) .

Q16:

Find the first derivative of π ( π₯ ) = οΉ 5 π₯ + 3 π₯ β 1 ο οΉ 9 π₯ + 5 π₯ + 2 ο 2 2 at ( β 1 , 6 ) .

Q17:

Find the first derivative of π ( π₯ ) = οΉ 2 π₯ β 5 π₯ β 3 ο οΉ 5 π₯ β π₯ + 2 ο 2 2 at ( 0 , β 6 ) .

Q18:

Given π¦ π¦ = β 6 π οΉ β π₯ + 3 ο : l n 4 π₯ 2 , determine d d π¦ π₯ .

Q19:

Determine the derivative of π¦ = π π₯ β 5 π₯ 2 .

Q20:

Find the first derivative of π ( π₯ ) = 2 π₯ ( π₯ β 3 ) ( π₯ β 1 ) ( π₯ + 2 ) at ( β 1 , β 1 6 ) .

Q21:

Find the first derivative of π ( π₯ ) = π₯ ( π₯ β 1 ) ( π₯ + 4 ) ( π₯ + 6 ) at ( β 5 , β 3 0 ) .

Q22:

If π₯ = 3 π¦ π 7 9 β π¦ , find d d π₯ π¦ .

Q23:

Given π¦ = 8 π 9 π₯ + 3 π₯ 7 π₯ 3 l n , find d d π¦ π₯ .

Q24:

Differentiate π ( π§ ) = 7 4 π§ π§ 7 l o g .

Q25:

Find the first derivative of π ( π₯ ) = οΉ π₯ + 4 ο οΊ 3 π₯ β π₯ β 7 ο οΊ 3 π₯ β π₯ + 7 ο 8 at π₯ = β 1 .

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