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Find the derivative of the following functions using the limit definition: i) y = x3 ii) y = 3x + 2

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2 parts
i

f′(x)=3x2f'(x) = 3x^2

ii

f′(x)=3f'(x) = 3

Step-by-step solution

7 steps

You're on the right track! Let's complete the first part and then move to the next.

Part i· 3 steps

y=x3y = x^3

  1. Calculate the difference f(x+h)−f(x)f(x+h) - f(x).

    f(x+h)−f(x)=(x3+3x2h+3xh2+h3)−x3=3x2h+3xh2+h3f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3) - x^3 = 3x^2h + 3xh^2 + h^3

  2. Substitute into the limit definition and simplify.

    f′(x)=lim⁡h→03x2h+3xh2+h3hf'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} f′(x)=lim⁡h→0h(3x2+3xh+h2)hf'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2)}{h} f′(x)=lim⁡h→0(3x2+3xh+h2)f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2)

  3. Evaluate the limit.

    f′(x)=3x2+3x(0)+(0)2f'(x) = 3x^2 + 3x(0) + (0)^2 f′(x)=3x2\boxed{f'(x) = 3x^2}

Part ii

f′(x)=3f'(x) = 3

y=3x+2y = 3x + 2

  1. Identify f(x)f(x) and f(x+h)f(x+h).

    f(x)=3x+2f(x) = 3x + 2 f(x+h)=3(x+h)+2=3x+3h+2f(x+h) = 3(x+h) + 2 = 3x + 3h + 2

  2. Calculate the difference f(x+h)−f(x)f(x+h) - f(x).

    f(x+h)−f(x)=(3x+3h+2)−(3x+2)=3hf(x+h) - f(x) = (3x + 3h + 2) - (3x + 2) = 3h

  3. Substitute into the limit definition and simplify.

    f′(x)=lim⁡h→03hhf'(x) = \lim_{h \to 0} \frac{3h}{h} f′(x)=lim⁡h→03f'(x) = \lim_{h \to 0} 3

  4. Evaluate the limit.

    f′(x)=3\boxed{f'(x) = 3}
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