Find the derivatives
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You need to know and apply the product rule, chain rule and quotient rule.
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Product, quotient, & chain rules | Khan Academy
Covered basic differentiation? Great! Now let's take things to the next level. In this topic, you will learn general rules that tell us how to differentiate products of functions, quotients of functions, and composite functions. Anxious to find the derivative of eˣ⋅sin(x²)? You've come to the...
HallsofIvy
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\(\displaystyle f(x)= [sin(4x+ 5)][ln(2x^2+ 5x- 1)]^3\)
The product rule says that the derivative of u times v is (uv)'= u'v+ uv'.
f is the product of sin(4x+ 5) and [ln(2x^2+ 5x- 1)]^3 so u= sin(4x+ 5) and v= [ln(2x^2+ 5x- 1)]^3. What is the derivative of sin(4x+ 5)? What is the derifative of [ln(2x^2+ 5x- 1)]^3?
(For the second, think of it as w^3 where w(x)= ln(2x^2+ 5x- 1). What is the derivative of w^3 with respect to w? What is the derivative of w(x)= ln(2x^2+ 5x- 1) with respect to x? Use the "chain rule".
The product rule says that the derivative of u times v is (uv)'= u'v+ uv'.
f is the product of sin(4x+ 5) and [ln(2x^2+ 5x- 1)]^3 so u= sin(4x+ 5) and v= [ln(2x^2+ 5x- 1)]^3. What is the derivative of sin(4x+ 5)? What is the derifative of [ln(2x^2+ 5x- 1)]^3?
(For the second, think of it as w^3 where w(x)= ln(2x^2+ 5x- 1). What is the derivative of w^3 with respect to w? What is the derivative of w(x)= ln(2x^2+ 5x- 1) with respect to x? Use the "chain rule".