Find the derivative using first principle

Mathematics

ScanSolve AI Answer

Answer

2x

Step-by-step solution

8 steps

  1. Define the function.

    Let f(x)=x2f(x) = x^2.

  2. Write down the formula for the derivative using the first principle.

    f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

  3. Find f(x+h)f(x+h).

    Substitute (x+h)(x+h) into the function f(x)f(x):

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

    Expand (x+h)2(x+h)^2:

    (x+h)2=x2+2xh+h2(x+h)^2 = x^2 + 2xh + h^2

  4. Substitute f(x+h)f(x+h) and f(x)f(x) into the first principle formula.

    f′(x)=lim⁡h→0(x2+2xh+h2)−x2hf'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2) - x^2}{h}

  5. Simplify the numerator.

    f′(x)=lim⁡h→02xh+h2hf'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}

  6. Factor out hh from the numerator.

    f′(x)=lim⁡h→0h(2x+h)hf'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}

  7. Cancel out hh (since h≠0h \neq 0 as h→0h \to 0).

    f′(x)=lim⁡h→0(2x+h)f'(x) = \lim_{h \to 0} (2x + h)

  8. Evaluate the limit by substituting h=0h=0.

    f′(x)=2x+0f'(x) = 2x + 0 f′(x)=2xf'(x) = 2x

    The derivative of x2x^2 using the first principle is:

    2x2x
Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QTo determine the derivative of the functions from first principles, we use the definition:QFind the second derivativeQFind the first derivative of the function.QState the definition of the derivative from first principles.QFind the standard derivative
Browse by topic:Calculus
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp