Math Problem Statement

Differentiate the following functions: 1) y = 12 2) y = x^7 3) y = 1/x^5 4) y = cube root of x 5) y = 9 times square root of x 6) y = 9t + 11 7) y = -3t^2 + 2t - 4 8) y = x(3x^2 - 9) 9) y = (4x^3 - 7x + 8) / x 10) y = (x^5 - 5x^3 + 2x) / x^3

Solution

The image you uploaded contains several differentiation problems. I'll guide you through the solutions step by step for each of them.

The goal is to differentiate the given functions with respect to the variable in each case.

1. y=12y = 12

The derivative of a constant is zero.

dydx=0\frac{dy}{dx} = 0


2. y=x7y = x^7

Use the power rule: ddx(xn)=nxn1\frac{d}{dx}(x^n) = n x^{n-1}.

dydx=7x6\frac{dy}{dx} = 7x^6


3. y=1x5y = \frac{1}{x^5}

Rewrite this as y=x5y = x^{-5}, then apply the power rule.

dydx=5x6=5x6\frac{dy}{dx} = -5x^{-6} = -\frac{5}{x^6}


4. y=x3y = \sqrt[3]{x}

Rewrite this as y=x13y = x^{\frac{1}{3}}, and apply the power rule.

dydx=13x23\frac{dy}{dx} = \frac{1}{3}x^{-\frac{2}{3}}


5. y=9xy = 9\sqrt{x}

Rewrite this as y=9x12y = 9x^{\frac{1}{2}}, and apply the power rule.

dydx=912x12=92x12=92x\frac{dy}{dx} = 9 \cdot \frac{1}{2}x^{-\frac{1}{2}} = \frac{9}{2}x^{-\frac{1}{2}} = \frac{9}{2\sqrt{x}}


6. y=9t+11y = 9t + 11

Differentiate term by term with respect to tt.

dydt=9\frac{dy}{dt} = 9


7. y=3t2+2t4y = -3t^2 + 2t - 4

Differentiate term by term.

dydt=6t+2\frac{dy}{dt} = -6t + 2


8. y=x(3x29)y = x(3x^2 - 9)

First, expand the expression: y=3x39xy = 3x^3 - 9x Now differentiate term by term.

dydx=9x29\frac{dy}{dx} = 9x^2 - 9


9. y=4x37x+8xy = \frac{4x^3 - 7x + 8}{x}

First, simplify the expression by dividing each term by xx: y=4x27+8x=4x27+8x1y = 4x^2 - 7 + \frac{8}{x} = 4x^2 - 7 + 8x^{-1} Now differentiate term by term.

dydx=8x8x2=8x8x2\frac{dy}{dx} = 8x - 8x^{-2} = 8x - \frac{8}{x^2}


10. y=x55x3+2xx3y = \frac{x^5 - 5x^3 + 2x}{x^3}

Simplify the expression by dividing each term by x3x^3: y=x25+2x2=x25+2x2y = x^2 - 5 + \frac{2}{x^2} = x^2 - 5 + 2x^{-2} Now differentiate term by term.

dydx=2x+4x3=2x+4x3\frac{dy}{dx} = 2x + 4x^{-3} = 2x + \frac{4}{x^3}


If you have any questions or need further details on any of the steps, feel free to ask!

Relative Questions:

  1. How does the power rule apply to negative exponents?
  2. What are the general rules for differentiating constant terms?
  3. How can we simplify expressions before differentiating them, especially fractions?
  4. How would you differentiate a product of two functions using the product rule?
  5. What steps should be taken when differentiating functions with radicals?

Tip:

Always simplify the function as much as possible before applying differentiation rules to make the process easier.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Power Rule
Constant Rule
Radical Functions
Polynomial Functions
Product Rule
Quotient Rule

Formulas

Power Rule: d/dx(x^n) = n*x^(n-1)
Constant Rule: d/dx(C) = 0
Derivative of x^n: d/dx(x^n) = n*x^(n-1)

Theorems

Power Rule
Quotient Rule
Product Rule
Radical Differentiation

Suitable Grade Level

Grades 11-12