Math Problem Statement

Find the first derivative of each algebraic function: 1. y = 3x^5 - 14x^3 + 12x; 2. y = √(1 + √(2 - x)); 3. y = (x + 1)/(x + 1); 4. y = (x + 3)(2x^2 - 11).

Solution

The image contains a set of algebraic functions for which you are asked to find the first derivative. Let's solve each problem step by step:

1. y=3x514x3+12xy = 3x^5 - 14x^3 + 12x

To find the first derivative, use the power rule ddxxn=nxn1\frac{d}{dx} x^n = nx^{n-1}.

dydx=ddx(3x5)ddx(14x3)+ddx(12x)\frac{dy}{dx} = \frac{d}{dx}(3x^5) - \frac{d}{dx}(14x^3) + \frac{d}{dx}(12x) =53x4314x2+12= 5 \cdot 3x^{4} - 3 \cdot 14x^{2} + 12 dydx=15x442x2+12\frac{dy}{dx} = 15x^4 - 42x^2 + 12


2. y=1+2xy = \sqrt{1 + \sqrt{2 - x}}

Here, we need to use the chain rule. Let's break it down step by step.

Let u=2xu = \sqrt{2 - x}, so y=1+uy = \sqrt{1 + u}.

First, differentiate yy with respect to uu: dydu=121+u\frac{dy}{du} = \frac{1}{2\sqrt{1 + u}}

Next, differentiate uu with respect to xx: dudx=ddx(2x)=122x\frac{du}{dx} = \frac{d}{dx}(\sqrt{2 - x}) = \frac{-1}{2\sqrt{2 - x}}

Now, apply the chain rule: dydx=dydududx=121+2x122x\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{2\sqrt{1 + \sqrt{2 - x}}} \cdot \frac{-1}{2\sqrt{2 - x}}

Thus, the derivative is: dydx=141+2x2x\frac{dy}{dx} = \frac{-1}{4\sqrt{1 + \sqrt{2 - x}} \cdot \sqrt{2 - x}}


3. y=x+1x+1y = \frac{x + 1}{x + 1}

This simplifies to y=1y = 1 since the numerator and denominator are the same, except for the case when x=1x = -1, where it is undefined.

For y=1y = 1, the derivative is: dydx=0\frac{dy}{dx} = 0


4. y=(x+3)(2x211)y = (x + 3)(2x^2 - 11)

Use the product rule ddx[uv]=uv+uv\frac{d}{dx} [uv] = u'v + uv', where u=(x+3)u = (x + 3) and v=(2x211)v = (2x^2 - 11).

First, find uu' and vv': u=ddx(x+3)=1u' = \frac{d}{dx}(x + 3) = 1 v=ddx(2x211)=4xv' = \frac{d}{dx}(2x^2 - 11) = 4x

Now apply the product rule: dydx=(1)(2x211)+(x+3)(4x)\frac{dy}{dx} = (1)(2x^2 - 11) + (x + 3)(4x) =2x211+4x(x+3)= 2x^2 - 11 + 4x(x + 3) =2x211+4x2+12x= 2x^2 - 11 + 4x^2 + 12x dydx=6x2+12x11\frac{dy}{dx} = 6x^2 + 12x - 11


Do you have any questions or would you like further clarification on any of these solutions?

5 Related Questions:

  1. What is the chain rule, and how is it applied to composite functions?
  2. How does the power rule for derivatives apply to terms like xnx^n?
  3. Can you explain the product rule and when it's necessary?
  4. How do you find the derivative of a square root function?
  5. Why does the derivative of a constant function equal zero?

Tip:

When dealing with products and compositions of functions, always consider whether you need to use the product rule, chain rule, or both together!

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Math Problem Analysis

Mathematical Concepts

Derivatives
Power Rule
Chain Rule
Product Rule

Formulas

Power Rule: d/dx(x^n) = n*x^(n-1)
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Product Rule: d/dx[uv] = u'v + uv'

Theorems

Chain Rule
Product Rule

Suitable Grade Level

Undergraduate