# Mixed Differentiation Problems 2 Answers 6 - 10

We assume you are familiar enough with the basics, Multiple Rule and Sum and Difference Rules to no longer need to mention which is being used when several are used with simple expressions such as polynomials.

1. y = ln cos 5x
2. This problem is a composite function of a composite function so use the Chain Rule twice.

= (cos 5x)      using the Chain Rule

= (−sin 5x)5x     using the Chain Rule

= (−sin 5x)(5)     using basic derivatives

= −5tan 5x     simplifying

3. y = 1 − x3e−9x
4. This problem simplifies to differentiating a product of a basic function and a composite function, so use the Product Rule then the Chain Rule.

= e−9x(−x3) + (−x3) e−9x     using the Product Rule

= e−9x(−x3) + (−x3) e−9x (−9x)    using the Chain Rule on the right hand expression

= e−9x(−3x2) + (−x3) e−9x (−9)    using basic derivatives

= (9x3 − 3x2) e−9x     tidying up

5. y =
6. This problem is a quotient of a composite function and a polynomial so the Quotient Rule and Chain Rule should be used.

=      using the Quotient Rule

=      using the Chain Rule on the left expression

=      using basic derivatives

=      simplifying

=      taking out the common factor of (3x − 5)2

=      multiplying out the right hand brackets

=      collecting like terms

7. y =
8. This problem is a variable power function so take the natural logarithm first then differentiateusing the Chain Rule.

ln y = (ln 6)     taking logs

= ½x−½(ln 6)     differentiating

= ½yx−½(ln 6)     multiplying both sides by y

=      substituting for y and expressing as a fraction

9. y = (1 + 2x)(3 − 4x)(5 + 6x)
10. This problem is a product of 3 polynomials. You can either use algebraic manipulation to expand the brackets to obtain one cubic polynomial then differentiate or you can use the Product Rule twice if you like a lot of work.

Either

y = (1 + 2x)(3 − 4x)(5 + 6x) = (3 + 6 x − 8x − 8x2)(5 + 6x) = (3 + 2x − 8x2)(5 + 6x)

y = 15 + 18x + 10x + 12x2 − 40x2 − 48x3 = 15 + 28x − 28x2 − 48x3    expanding the brackets

= 28 − 56x − 144x2

Or

= (5 + 6x) (1 + 2x)(3 − 4x) + (1 + 2x)(3 − 4x)(5 + 6x)     applying the Product Rule

= (5 + 6x)((3 − 4x) (1 + 2x) + (1 + 2x) (3 − 4x)) + (1 + 2x)(3 − 4x)(5 + 6x)     applying the Product Rule to the left expression

= (5 + 6x)((3 − 4x) (2) + (1 + 2x)(−4)) + (1 + 2x)(3 − 4x)(6)     using basic derivatives

= (5 + 6x)(6 − 8x − 4 − 8x) + 6(1 + 2x)(3 − 4x)     expanding brackets on the left

= (5 + 6x)(2 − 16 x) + 6(1 + 2x)(3 − 4x)     collecting like terms on the left

= (10 + 12x − 80x − 96x2) + 6(3 − 4x + 6x − 8x2)     expanding brackets

= (10 − 68x − 96x2) + 6(3 + 2x − 8x2    collecting like terms

= (10 − 68x − 96x2) + (18 + 12x − 48x2)     expanding bracket

= 28 − 56 x − 144x2     collecting like terms