Implicit differentiation of 5x^2-2xy+7^y
Two assumptions
1. you want dy/dx
2. You have the equation 5x^2 - 2xy + 7y^2 = 0
10x - 2x dy/dx - 2y + 14y dy/dx = 0
dy/dx( 14y - 2x) = 2y - 10x
dy/dx = (2y-10x)/(14y - 2x) = (y - 5x)/(7y - x)
No, the equation is
5x^2-2xy+7^y
Last term has 7 to the y power....
Ouch, nasty term ...
the derivative of 7^y would be
(ln7)(7^y)dy/dx
I am sure you can make the necessary changes .
Yes...very nasty! Thanks
To implicitly differentiate the expression 5x^2 - 2xy + 7^y, we can follow these steps:
Step 1: Treat all variables as functions of either x or y.
Let's assume that x = x(x) and y = y(x), meaning x is a function of x and y is a function of x.
Step 2: Apply the chain rule for differentiating the y terms.
Differentiating y with respect to x, we get dy/dx.
Step 3: Determine the derivative of each term with respect to x.
Let's differentiate each term separately:
- The derivative of 5x^2 is 10x.
- The derivative of -2xy is -2(x * dy/dx + y).
Here, we used the product rule: the derivative of the first term (-2x) times the derivative of the second term (dy/dx) plus the derivative of the second term (y) times the derivative of the first term (-2x).
- The derivative of 7^y is (7^y) * ln(7) * dy/dx.
We used the chain rule here, as the derivative of 7^y with respect to y is (7^y) * ln(7) and then multiplied it by dy/dx.
Step 4: Write the entire equation in terms of dy/dx.
Combining all the derivatives obtained in step 3, we get:
10x - 2(x * dy/dx + y) + (7^y) * ln(7) * dy/dx = 0.
Step 5: Solve for dy/dx.
Now we need to isolate dy/dx to find its value. Let's rearrange the equation:
10x - 2xy - 2y + (7^y) * ln(7) * dy/dx = 0.
To isolate dy/dx, we move the two terms involving dy/dx to the left side:
(7^y) * ln(7) * dy/dx - 2xy + 2y = -10x.
Finally, we divide both sides by [(7^y) * ln(7)] to obtain the value of dy/dx:
dy/dx = (-10x + 2xy - 2y) / [(7^y) * ln(7)].
And that's the implicit differentiation of the expression 5x^2 - 2xy + 7^y!