If 3x^2+3x+xy=4 and y(4)=−14,
find y'(4) by implicit differentiation.
3x^2+3x+xy=4
dy/dx (3x^2+3x+xy) = dy/dx (4)
dy/dx (3x^2) + dy/dx (3x) + dy/dx (xy) = 0
6x+3+x(dy/dx)+y = 0
x(dy/dx) = -6x-3-y
dy/dx = (-6x-3-y)/x
y'(4) = (-6(4)-3-(-14))/(4)
y'(4) = (-24-3-(-14))/4
y'(4) = (-27+14)/4
y'(4) = -13/4
6 x dx + 3 dx + x dy +y dx = 0
(6 x + 3 + y ) dx + x dy = 0
x dy/dx = -( 6x + 3 + y)
at x = 4, y = -14
so at x = 4
4 dy/dx =- ( 24 + 3 - 14 ) = -13
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3x^2+3x+xy=4
In recrange Evaluate at (x0,y0) type 4 and -14
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You will see solution step-by-step
To find the derivative of y with respect to x, we will differentiate both sides of the given equation with respect to x while treating y as a function of x. This is known as implicit differentiation.
Let's go step by step:
Step 1: Differentiate each term with respect to x.
The first term, 3x^2, differentiates to 6x.
The second term, 3x, differentiates to 3.
The third term, xy, requires the product rule. The derivative of xy with respect to x is given by:
(derivative of x) * y + x * (derivative of y)
So, the derivative of xy with respect to x is y + x(y').
Step 2: Set up the equation in terms of y' and solve for it.
After differentiating each term, we have:
6x + 3 + y + x(y') = 0
Step 3: Simplify the equation.
Rearrange the equation to isolate y':
y + x(y') = -6x - 3
y' = (-6x - 3 - y) / x
Step 4: Substitute the given value.
We are asked to find y'(4) at x = 4, and y(4) is given as -14.
Substituting these values into the equation for y':
y'(4) = (-6(4) - 3 - (-14)) / 4
y'(4) = (-24 - 3 + 14) / 4
y'(4) = -13 / 4
Therefore, y'(4) = -13 / 4 is the answer to your question.