Question

if y=4x^2, find the slope of y-x curve at a point P(1,4)

Answer 1

If y = 4x² then the slope of y-x curve at a point P(1,4) is 8

Given :

The curve y = 4x²

To find :

The slope of the y-x curve at a point P(1,4)

Solution :

Step 1 of 2 :

Write down the given curve

Here the given equation of the curve is

y = 4x²

Step 2 of 2 :

Find slope of y-x curve at a point P(1,4)

$\displaystyle \sf{ y = 4 {x}^{2} }$

Differentiating both sides with respect to x we get

$\displaystyle \sf{ \frac{dy}{dx} = \frac{d}{dx}(4 {x}^{2}) }$

$\displaystyle \sf{ \implies \frac{dy}{dx} = 4\frac{d}{dx}( {x}^{2})}$

$\displaystyle \sf{ \implies \frac{dy}{dx} = 4 \times 2x}$

$\displaystyle \sf{ \implies \frac{dy}{dx} = 8x}$

Hence the slope of the y-x curve at a point P(1,4)

$\displaystyle \sf{ = \frac{dy}{dx} \bigg|_{P(1,4)} }$

$\displaystyle \sf{ = 8 \times 1 }$

$= 8$

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Answer 2

For the given curve $y=4 x^{2}$  the slope of $y-x$curve at a point $P(1,4)$ is 8

Curve

A curve is defined as a smoothly- flowing continuous line that has bent. It does not have any sharp turns. The way to identify the curve is that the line bends and changes its direction at least once.

Slope of a curve:

The slope of a curve at a point is equal to the slope of the straight line that is tangent to the curve at that point.

Given :

The curve $y=4 x^{2}$

Step by step solution

Here the given equation of the curve is

$y=4 x^{2}$

Differentiating both sides with respect to $x$

$&\frac{d y}{d x}=\frac{d}{d x}\left(4 x^{2}\right) \\&\Longrightarrow \frac{d y}{d x}=4 \frac{d}{d x}\left(x^{2}\right) \\&\Longrightarrow \frac{d y}{d x}=4 \times 2 x\end{aligned}$

$\Longrightarrow \frac{d y}{d x}=8 x......(1)$

Substitute the given point

Substitute the value of point $P(1,4)$ in equation (1).

$&=\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{\mathrm{P}(1,4)}$

$&=8 \times 1$

$&=8$

Hence the slope of the $y-x$ curve at a point $P(1,4)$ is 8

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