Math Problem Statement

Give me the answer of this question with explanation

Give me the answer of this question with explanation

Solution

Let’s solve these derivatives step by step. These involve applications of the chain rule.


(i) (5x+3)3(5x + 3)^{-3}

We use the chain rule here. Let u=5x+3u = 5x + 3, then y=u3y = u^{-3}.

  1. Derivative of u3u^{-3} is: dydu=3u4\frac{dy}{du} = -3u^{-4}

  2. Derivative of u=5x+3u = 5x + 3 is: dudx=5\frac{du}{dx} = 5

Using the chain rule: dydx=dydududx=3(5x+3)45\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -3(5x + 3)^{-4} \cdot 5

Final result: dydx=15(5x+3)4\frac{dy}{dx} = -15(5x + 3)^{-4}


(ii) (23x)5(2 - 3x)^5

Again, applying the chain rule. Let u=23xu = 2 - 3x, then y=u5y = u^5.

  1. Derivative of u5u^5 is: dydu=5u4\frac{dy}{du} = 5u^4

  2. Derivative of u=23xu = 2 - 3x is: dudx=3\frac{du}{dx} = -3

Using the chain rule: dydx=dydududx=5(23x)4(3)\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5(2 - 3x)^4 \cdot (-3)

Final result: dydx=15(23x)4\frac{dy}{dx} = -15(2 - 3x)^4


(iii) (3+7x)5/3(3 + 7x)^{-5/3}

We proceed similarly. Let u=3+7xu = 3 + 7x, then y=u5/3y = u^{-5/3}.

  1. Derivative of u5/3u^{-5/3} is: dydu=53u8/3\frac{dy}{du} = -\frac{5}{3}u^{-8/3}

  2. Derivative of u=3+7xu = 3 + 7x is: dudx=7\frac{du}{dx} = 7

Using the chain rule: dydx=dydududx=53(3+7x)8/37\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\frac{5}{3}(3 + 7x)^{-8/3} \cdot 7

Final result: dydx=353(3+7x)8/3\frac{dy}{dx} = -\frac{35}{3}(3 + 7x)^{-8/3}


Final Answers:

  1. dydx=15(5x+3)4\frac{dy}{dx} = -15(5x + 3)^{-4}
  2. dydx=15(23x)4\frac{dy}{dx} = -15(2 - 3x)^4
  3. dydx=353(3+7x)8/3\frac{dy}{dx} = -\frac{35}{3}(3 + 7x)^{-8/3}

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Related Questions:

  1. What is the chain rule, and how is it applied in calculus?
  2. How do you differentiate (ax+b)n(ax + b)^n in general?
  3. What happens if nn is a fractional power, like in part (iii)?
  4. Can the product rule and chain rule ever be applied together?
  5. How would you compute second derivatives of these functions?

Tip:

Always simplify inner derivatives first before multiplying them with the outer derivative—it avoids mistakes in complex chain rule problems.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Chain Rule

Formulas

Derivative of a power function: d/dx[u^n] = n*u^(n-1) * du/dx
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Theorems

Chain Rule of Differentiation

Suitable Grade Level

Grades 11-12