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Sum

Find the derivative of the following function by the first principle: 3x^{2 }+ 4

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#### Solution

Let f(x) = 3x^{2} + 4

∴ f(x + h) = 3(x + h)^{2} + 4

= 3(x^{2} + 2xh + h^{2}) + 4

= 3x^{2} + 6xh + 3h^{2} + 4

By first principle, we get

f ′(x) =`lim_("h"→ 0) ("f"(x + "h") - "f"(x))/"h"`

=`lim_("h" → 0) ((3x^2 + 6x"h" + 3"h"^2 + 4) - (3x^2 + 4))/"h"`

=`lim_("h" → 0) (3"h"^2 + 6x"h")/"h"`

=`lim_("h"→0) (h(3h+6x))/h`

=`lim_("h" → 0)(6x + 3"h")` …[∵ h → 0, ∴h ≠ 0]

= 6x + 3(0)

= 6x

Concept: Rules of Differentiation (Without Proof)

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