Math Problem Statement

Find f'(x) for each of the given functions: a) f(x) = 2x + 1/4 x^6, b) f(x) = x^4 - 5, c) f(x) = x + 4x^(1/2), d) f(x) = 6x^(5/2) - x^(-4), e) f(x) = 7 + x^3, f) f(x) = 2x^4 + x^2, g) f(x) = 3x^4 - 5x^3, h) f(x) = 2 - 7x^(-1) + x^3.

Solution

The image asks to find the derivative f(x)f'(x) for each of the given functions. Let's go through each part:

a) f(x)=2x+14x6f(x) = 2x + \frac{1}{4}x^6

  • The derivative of 2x2x is 22.
  • The derivative of 14x6\frac{1}{4}x^6 is 64x5=32x5\frac{6}{4}x^5 = \frac{3}{2}x^5. Thus, f(x)=2+32x5f'(x) = 2 + \frac{3}{2}x^5.

b) f(x)=x45f(x) = x^4 - 5

  • The derivative of x4x^4 is 4x34x^3.
  • The derivative of the constant 5-5 is 0. Thus, f(x)=4x3f'(x) = 4x^3.

c) f(x)=x+4x12f(x) = x + 4x^{\frac{1}{2}}

  • The derivative of xx is 11.
  • The derivative of 4x124x^{\frac{1}{2}} is 412x12=2x124 \cdot \frac{1}{2}x^{-\frac{1}{2}} = 2x^{-\frac{1}{2}}. Thus, f(x)=1+2x12f'(x) = 1 + 2x^{-\frac{1}{2}}.

d) f(x)=6x52x4f(x) = 6x^{\frac{5}{2}} - x^{-4}

  • The derivative of 6x526x^{\frac{5}{2}} is 652x32=15x326 \cdot \frac{5}{2}x^{\frac{3}{2}} = 15x^{\frac{3}{2}}.
  • The derivative of x4-x^{-4} is (4)x5=4x5-(-4)x^{-5} = 4x^{-5}. Thus, f(x)=15x32+4x5f'(x) = 15x^{\frac{3}{2}} + 4x^{-5}.

e) f(x)=7+x3f(x) = 7 + x^3

  • The derivative of 77 is 0.
  • The derivative of x3x^3 is 3x23x^2. Thus, f(x)=3x2f'(x) = 3x^2.

f) f(x)=2x4+x2f(x) = 2x^4 + x^2

  • The derivative of 2x42x^4 is 8x38x^3.
  • The derivative of x2x^2 is 2x2x. Thus, f(x)=8x3+2xf'(x) = 8x^3 + 2x.

g) f(x)=3x45x3f(x) = 3x^4 - 5x^3

  • The derivative of 3x43x^4 is 12x312x^3.
  • The derivative of 5x3-5x^3 is 15x2-15x^2. Thus, f(x)=12x315x2f'(x) = 12x^3 - 15x^2.

h) f(x)=27x1+x3f(x) = 2 - 7x^{-1} + x^3

  • The derivative of 22 is 0.
  • The derivative of 7x1-7x^{-1} is 7x27x^{-2}.
  • The derivative of x3x^3 is 3x23x^2. Thus, f(x)=7x2+3x2f'(x) = 7x^{-2} + 3x^2.

Would you like further details or clarifications on any of these derivatives?

Here are some related questions you might want to explore:

  1. What is the second derivative of the function in part (a)?
  2. How do higher powers of xx affect the shape of the derivative graph?
  3. Can we apply chain rule in these examples? If so, in what cases?
  4. How do we differentiate functions with multiple terms like in part (h)?
  5. What is the physical interpretation of these derivatives in terms of rates of change?

Tip: Remember, constants always differentiate to zero, and powers of xx follow the power rule for differentiation!

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Power Rule

Formulas

d/dx (x^n) = n * x^(n-1)

Theorems

Power Rule for Derivatives

Suitable Grade Level

Grades 11-12