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What is the slope of tangent line to y=3x^2+9 at x=2 ?

Option: 1

12


Option: 2

6


Option: 3

3


Option: 4

0


Answers (1)

best_answer

 

 

DIFFERENTIATION -

DIFFERENTIATION

Derivative is the rate of change of a function with respect to dependent variable at any instant. Also, the derivative tells the slope of tangent of the function at that point.
 

\mathrm{\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\frac{\mathit{d} f(x)}{\mathit{d} x}=\tan\theta}.

Here, d f(x) is is very small change in value of f(x) produced by very small change of dx in the value of x.

The ratio \frac{\mathit{d} f(x)}{\mathit{d} x} is called the derivative of f(x) wwith respect to x and  tan ? is the slope of the tangent to the curve at point P.

The ratio\frac{\mathit{d} f(x)}{\mathit{d} x}\;\;\text{or}\;\;\frac{d}{dx}f(x) also called differential coefficient, also it is denoted by f’(x).

If we replace Δx with h, then we can write

\\\mathrm{\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}} \\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\lim_{\mathit{h} \rightarrow 0}\frac{f(x+\mathit{h})-f(x)}{\mathit{h}}=\frac{\mathit{dy}}{\mathit{dx}}=\mathit{f'(x)}}

This is also known as the derivative of f(x) from the first principle.

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y=3x^2+9\\ \frac{dy}{dx}=6x\\ slope=\frac{dy}{dx}|_{x=2}=6 \times 2=12\\

Posted by

Suraj Bhandari

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