Let's proceed step by step:
1. Differentiating Implicitly:
- We start by differentiating the given equation, 9x^2 - 4y - 17 = 0, with respect to 'x'.
- The derivative of 9x^2 with respect to x is (18x) and the derivative of -4y with respect to x is (-4 times derivative of y with respect to x) and the derivative of 17 with respect to x is 0.
- Therefore, the resultant expression for the derivative of the equation with respect to x is 18x - 4(dy/dx)=0.
2. Solving for y′(dy/dx):
- Solve the expression found in step 1 to find y'.
- In this case, there is no explicit y'. This occurs when y' cannot be directly isolated from the differentiated equation.
3. Solve for y:
- Next, we solve the original equation for y.
- We rearrange the given equation to solve for y and obtain y = (9x^2/4) - (17/4)
4. Differentiating directly:
- Finally, differentiate the expression for y with respect to x from step 3.
- Differentiating (9x^2/4) with respect to x gives us 9x/2.
- And since 17/4 is a constant, its derivative equals zero.
Therefore, the derivative of the original equation after implicit differentiation is 18x. There is no explicit y', the equation for y after solving from the original equation is y = (9x^2/4) - (17/4) and the derivative of y with respect to x is 9x/2.
the derivative of y= (9x^2/4) - (17/4) with respect to x is 9x/2.