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.

    dy by dx = 5d by dx(cos 5x)      using the Chain Rule

    dy by dx = 5(−sin 5x)d by dx5x     using the Chain Rule

    dy by dx = 5(−sin 5x)(5)     using basic derivatives

    dy by dx = −5tan 5x     simplifying

  2. y = 1 − x3e−9x

    3. This problem simplifies to differentiating a product of a basic function and a composite function, so use the Product Rule then the Chain Rule.

    dy by dx = e−9xd by dx(−x3) + (−x3) d by dxe−9x     using the Product Rule

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

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

    dy by dx = (9x3 − 3x2) e−9x     tidying up

  3. y = (3x minus 5) to the power 3 over (2x plus 1)

    4. This problem is a quotient of a composite function and a polynomial so the Quotient Rule and Chain Rule should be used.

    dy by dx = (2x + 1) times the derivative of (3x - 5) cubed minus (3x - 5) cubed times the derivative of (2x + 1) all divided by (2x + 1) squared     using the Quotient Rule

    dy by dx = (2x + 1) times 3 times (3x - 5) squared times the derivative of (3x - 5) minus (3x - 5) cubed times the derivative of (2x + 1) all divided by (2x + 1) squared     using the Chain Rule on the left expression

    dy by dx = ((2x + 1) times 3 times (3x - 5) squared times 3) minus ((3x - 5) cubed times 2) all divided by (2x + 1) squared     using basic derivatives

    dy by dx = ((2x + 1) times 9 times (3x - 5) squared) minus 2 times ((3x - 5) cubed) all divided by (2x + 1) squared     simplifying

    dy by dx = ((3x - 5) squared) times ((2x + 1) times 9 minus 2 times (3x - 5)) all divided by (2x + 1) squared     taking out the common factor of (3x − 5)2

    dy by dx = (3x - 5) squared) times (18x + 9 - 6x + 10) all divided by (2x + 1) squared     multiplying out the right hand brackets

    dy by dx = (3x - 5) squared) times (12x + 19) all divided by (2x + 1) squared     collecting like terms

  4. y = 6 to the power of the square root of x

    5. This problem is a variable power function so take the natural logarithm first then differentiateusing the Chain Rule.

    ln y = square root of x(ln 6)     taking logs

    1 over ydy by dx = ½x−½(ln 6)     differentiating

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

    dy by dx = ln 6 times 6 to the power of the square root of x all over 2 square root of x     substituting for y and expressing as a fraction

  5. y = (1 + 2x)(3 − 4x)(5 + 6x)

    6. 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

    dy by dx = 28 − 56x − 144x2

    Or

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

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

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

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

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

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

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

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

    dy by dx = 28 − 56 x − 144x2     collecting like terms

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