Solve the questions in a simple matter but must show work. Find the derivative of the function. 1)y = x3 x – 1 1) 2)y = x + 1 x x – 1 x 2) 3)y = (-3x + 7)5 3) 4)q = 20r – r7 4)

QUESTION

Solve the questions in a simple matter but must show work.

Find the derivative of the function. 1)y = x3 x – 1 1)

2)y = x + 1 x x – 1 x 2)

3)y = (-3x + 7)5 3)

4)q = 20r – r7 4)

1

5)y = ln(x – 3) 5)

6)y = ln x x5 6)

7)f(x) = 8e12x 7)

8)h(x) = ex 8×2 + 7 8)

2

Provide an appropriate response. 9)Find an equation for the tangent to the curve y = 27 x2 + 2 at the point (1, 9). 9)

Find all values of x for the given function where the tangent line is horizontal. 10)f(x) = x (x2 + 3)3 10)

Find the future value. 11)$1972 invested for 12 years at 4% compounded quarterly 11)

Find the present value of the future value. 12)$11,000, invested for 4 years at 3% compounded continuously 12)

3

Solve the problem. 13)Find the required annual interest rate, to the nearest tenth of a percent, for $1100 to grow to $1400 if interest is compounded continuously

13)

14)How long will it take money to double if it is invested at 8%, compounded continuously? Round your answer to the nearest tenth.

14)

Solve the problem. 15)The concentration of a certain drug in the bloodstream t minutes after swallowing a pill containing the drug can be approximated using the equation C(t) = 1 6 4t + 1 -1/2, where C(t) is the concentration in arbitrary units and t is in minutes. Find the rate of change of concentration with respect to time at t = 12 minutes and intrepret the results

ANSWER

Sure! Let’s solve the given problems step by step.

1) Find the derivative of the function:

1) y = x^3(x – 1)

To find the derivative of this function, we can apply the product rule.

Let’s differentiate each term separately:

dy/dx = d(x^3)/dx * (x – 1) + x^3 * d(x – 1)/dx

dy/dx = 3x^2 * (x – 1) + x^3 * 1

Simplifying further:

dy/dx = 3x^3 – 3x^2 + x^3

dy/dx = 4x^3 – 3x^2

Therefore, the derivative of the function y = x^3(x – 1) is dy/dx = 4x^3 – 3x^2.

2) y = (x + 1)(x – 1)(x^2)

To find the derivative, we can use the product rule and the power rule.

dy/dx = (d(x + 1)/dx)(x – 1)(x^2) + (x + 1)(d(x – 1)/dx)(x^2) + (x + 1)(x – 1)(d(x^2)/dx)

dy/dx = (1)(x – 1)(x^2) + (x + 1)(1)(x^2) + (x + 1)(x – 1)(2x)

Simplifying further:

dy/dx = (x – 1)(x^2) + (x + 1)(x^2) + 2x(x + 1)(x – 1)

dy/dx = x^3 – x^2 + x^3 + x^2 + 2x^3 – 2x

dy/dx = 4x^3 – 2x

Therefore, the derivative of the function y = (x + 1)(x – 1)(x^2) is dy/dx = 4x^3 – 2x.

3) y = (-3x + 7)^5

To find the derivative, we can apply the chain rule and the power rule.

Let’s differentiate each term separately:

dy/dx = 5(-3x + 7)^4 * d(-3x + 7)/dx

dy/dx = 5(-3x + 7)^4 * (-3)

Simplifying further:

dy/dx = -15(-3x + 7)^4

Therefore, the derivative of the function y = (-3x + 7)^5 is dy/dx = -15(-3x + 7)^4.

4) q = 20r – r^7

To find the derivative, we can use the power rule.

dy/dx = d(20r)/dx – d(r^7)/dx

dy/dx = 20 * d(r)/dx – 7r^6

Simplifying further:

dy/dx = 20 – 7r^6

Therefore, the derivative of the function q = 20r – r^7 is dq/dr = 20 – 7r^6.

5) y = ln(x – 3)

To find the derivative, we can use the chain rule.

Let’s differentiate each term separately:

dy/dx = 1/(x – 3) * d(x – 3)/dx

dy/dx = 1/(x – 3) * 1

Simplifying further:

dy/dx = 1

/(x – 3)

Therefore, the derivative of the function y = ln(x – 3) is dy/dx = 1/(x – 3).

6) y = ln(x * x^5)

To find the derivative, we can use the product rule and the chain rule.

Let’s differentiate each term separately:

dy/dx = d(ln(x))/dx + d(ln(x^5))/dx

dy/dx = 1/x + 1/(x^5) * d(x^5)/dx

dy/dx = 1/x + 1/(x^5) * 5x^4

Simplifying further:

dy/dx = 1/x + 5/x

dy/dx = 6/x

Therefore, the derivative of the function y = ln(x * x^5) is dy/dx = 6/x.

7) f(x) = 8e^(12x)

To find the derivative, we can use the chain rule and the derivative of e^x.

Let’s differentiate each term separately:

df/dx = 8 * d(e^(12x))/dx

df/dx = 8 * 12e^(12x)

Simplifying further:

df/dx = 96e^(12x)

Therefore, the derivative of the function f(x) = 8e^(12x) is df/dx = 96e^(12x).

8) h(x) = e^(8x^2 + 7)

To find the derivative, we can use the chain rule and the derivative of e^x.

Let’s differentiate each term separately:

dh/dx = d(e^(8x^2 + 7))/dx

dh/dx = e^(8x^2 + 7) * d(8x^2 + 7)/dx

dh/dx = e^(8x^2 + 7) * (16x)

Simplifying further:

dh/dx = 16xe^(8x^2 + 7)

Therefore, the derivative of the function h(x) = e^(8x^2 + 7) is dh/dx = 16xe^(8x^2 + 7).

Let’s move on to the next set of questions.