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Derivative of:
Derivative of y=sqrt(2ax-x^2)
Derivative of y=ln(coshx)
Derivative of ax*e^(-ax^2)
Derivative of 3sin2x*cosx
Integral of d{x}:
sqrt(2ax-x^2)
Identical expressions
y=sqrt(two ax-x^2)
y equally square root of (2ax minus x squared )
y equally square root of (two ax minus x squared )
y=√(2ax-x^2)
y=sqrt(2ax-x2)
y=sqrt2ax-x2
y=sqrt(2ax-x²)
y=sqrt(2ax-x to the power of 2)
y=sqrt2ax-x^2
Similar expressions
y=sqrt(2ax+x^2)

The derivative of the function/ y=sqrt(2ax-x^2)

Derivative of y=sqrt(2ax-x^2)

The solution

You have entered [src]

              2
  /         2\ 
t*\2*a*x - x /

$$t \left(2 a x - x^{2}\right)^{2}$$

  /              2\
d |  /         2\ |
--\t*\2*a*x - x / /
dx

$$\frac{\partial}{\partial x} t \left(2 a x - x^{2}\right)^{2}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 2 a x - x^{2}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \left(2 a x - x^{2}\right)$ :
  1. Differentiate $2 a x - x^{2}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2 a$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $- 2 x$
    The result is: $2 a - 2 x$
  The result of the chain rule is:
  
  $\left(2 a - 2 x\right) \left(4 a x - 2 x^{2}\right)$
So, the result is: $t \left(2 a - 2 x\right) \left(4 a x - 2 x^{2}\right)$
Now simplify:

$4 t x \left(a - x\right) \left(2 a - x\right)$

The answer is:

$4 t x \left(a - x\right) \left(2 a - x\right)$

The first derivative [src]

               /         2\
t*(-4*x + 4*a)*\2*a*x - x /

$$t \left(4 a - 4 x\right) \left(2 a x - x^{2}\right)$$

Simplify

The second derivative [src]

    / 2            2        \
4*t*\x  + 2*(a - x)  - 2*a*x/

$$4 t \left(- 2 a x + x^{2} + 2 \left(a - x\right)^{2}\right)$$

Simplify

The third derivative [src]

-24*t*(a - x)

$$- 24 t \left(a - x\right)$$

Simplify