Differentiate, do not simplify.
a) y=(3x+2)(x^2-7)
b) x^3+3y^2=36
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derivative x^3+3y^2=36
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To differentiate the given equations without simplifying them, we can use the product rule and the chain rule for differentiation.
a) y = (3x + 2)(x^2 - 7)
To differentiate this equation, we will use the product rule, which states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
Let's label the two functions in our equation as f(x) = 3x + 2 and g(x) = x^2 - 7. Applying the product rule, we can differentiate y with respect to x as follows:
dy/dx = f'(x)g(x) + f(x)g'(x)
Now, let's find the derivatives of f(x) and g(x) separately.
The derivative of f(x) = 3x + 2 with respect to x is f'(x) = 3.
The derivative of g(x) = x^2 - 7 with respect to x is g'(x) = 2x.
Substituting these values into the product rule formula, we get:
dy/dx = (3)(x^2 - 7) + (3x + 2)(2x)
= 3x^2 - 21 + 6x^2 + 4x
= 9x^2 + 4x - 21
So, the derivative of y with respect to x is dy/dx = 9x^2 + 4x - 21.
b) x^3 + 3y^2 = 36
In this equation, we need to differentiate implicitly since both x and y are present in the equation.
Differentiating both sides of the equation with respect to x, we get:
d/dx (x^3) + d/dx (3y^2) = d/dx (36)
Using the power rule for differentiation, we find:
3x^2 + 6y(dy/dx) = 0
Now, let's isolate dy/dx to find the derivative of y with respect to x:
6y(dy/dx) = -3x^2
Dividing both sides by 6y, we get:
dy/dx = -x^2 / (2y)
Therefore, the derivative of y with respect to x is dy/dx = -x^2 / (2y).