Section 3.5 Implicit Differentiation. Example. If f ( x ) = ( x 7 + 3 x 5 – 2 x 2 ) 10 , determine f ’ (x) . Answer: f ΄ ( x ) = 10( x 7 + 3 x 5 – 2 x 2 ) 9 (7 x 6 + 15 x 4 – 4 x ). Now write the answer above only in terms of y if y = ( x 7 + 3 x 5 – 2 x 2 ) .
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If f(x) = (x7 + 3x5 – 2x2)10, determine f ’(x).
Answer:f΄(x)=10(x7 + 3x5 – 2x2)9(7x6 + 15x4 – 4x)
Now write the answer above only in terms of y if y = (x7 + 3x5 – 2x2).
If y is some unknown function of x, then
9x + x2– 2y = 5
5x – 3xy + y2 = 2y
Easy to solve for y and differentiate
Not easy to solve for y and differentiate
In equations like 5x – 3xy + y2 = 2y, we simply assume that y = f(x), or some function of x which is not easy to find.
Process wise, simply take the derivative of each side of the equation with respect to x and when we encounter terms containing y, we use the chain rule.
y3 = 2x
Solving for y’, we have the derivative
x2y3 = -7
Solving for y’, we have
Since y is a function of x, every time we differentiate a term containing y, we need to multiply it by y’or dy/dx
Determine dy/dx for the following.
Determine the first derivative of each of the following.
Inverse Trig Functions