A researcher believes that headaches can be relieved by drinking water flavored by

Bas_tet [7]

Use the power, product, and chain rules:

$y = x^2 (3x-1)^3$

• product rule

$\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\mathrm d(x^2)}{\mathrm dx}\times(3x-1)^3 + x^2\times\dfrac{\mathrm d(3x-1)^3}{\mathrm dx}$

• power rule for the first term, and power/chain rules for the second term:

$\dfrac{\mathrm dy}{\mathrm dx} = 2x\times(3x-1)^3 + x^2\times3(x-1)^2\times\dfrac{\mathrm d(3x-1)}{\mathrm dx}$

• power rule

$\dfrac{\mathrm dy}{\mathrm dx} = 2x\times(3x-1)^3 + x^2\times3(3x-1)^2\times3$

Now simplify.

$\dfrac{\mathrm dy}{\mathrm dx} = 2x(3x-1)^3 + 9x^2(3x-1)^2 \\\\ \dfrac{\mathrm dy}{\mathrm dx} = x(3x-1)^2 \times (2(3x-1) + 9x) \\\\ \boxed{\dfrac{\mathrm dy}{\mathrm dx} = x(3x-1)^2(15x-2)}$

You could also use logarithmic differentiation, which involves taking logarithms of both sides and differentiating with the chain rule.

On the right side, the logarithm of a product can be expanded as a sum of logarithms. Then use other properties of logarithms to simplify

$\ln(y) = \ln\left(x^2(3x-1)^3\right) \\\\ \ln(y) = \ln\left(x^2\right) + \ln\left((3x-1)^3\right) \\\\ \ln(y) = 2\ln(x) + 3\ln(3x-1)$

Differentiate both sides and you end up with the same derivative:

$\dfrac1y\dfrac{\mathrm dy}{\mathrm dx} = \dfrac2x + \dfrac9{3x-1} \\\\ \dfrac1y\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{15x-2}{x(3x-1)} \\\\ \dfrac{\mathrm dy}{\mathrm dx} = \dfrac{15x-2}{x(3x-1)} \times x^2(3x-1)^3 \\\\ \dfrac{\mathrm dy}{\mathrm dx} = x(15x-2)(3x-1)^2$

7 0

3 years ago

The area of a rectangle is found by multiplying its length and width together. Calculate the area, in square feet, of the room y

Vinil7 [7]

Answer:

Where is the question? Did you forget to attach the image, or did you need to create several possibilities?

Step-by-step explanation:

If you wanted several possibilities:

5 0

3 years ago

Write equivalent fractions for 3\5 and 1\4 using 20 as the conman denominator

Sindrei [870]

Answer:

12/20=3/5 5/20=1/4 so the answer would be 12/20 + 5/20

4 0

2 years ago

Can you help me please

Mekhanik [1.2K]

Answer:

-3

1.1 ×10

Explanation: when you divide it it comes to .0011

and with significant figures you move the decimal over until it is right after the first number that is not zero. so that would change .0011 to 1.1

also with significant figures if you move the decimal to the right you exponent by the 10 will be negative and if you move it to the left it will be positive. so I. this case we are moving g to the right 3 digits so our exponent will be negative 3.

and because the answer is still really. 0011 you multiply

-3

1.1 × 10

5 0

3 years ago

Can someone be my friend please!!!

KIM [24]

Answer:

no problem sure!

Step-by-step explanation:

;D

4 0

3 years ago