More Multiple Chain Rule Examples

1 Find the derivative of:

y equals the cosine of the exponent of (-2x)
y equals the cosine of u and u equals the exponent of (-2x)
u equals the exponent of v and v = -2x
dy by du equals the negative sine of u, du by dv equals the exponent of v and dv by dx equals -2
dy by dx equals dy by du times du by dv times dv by dx equals -2 times the exponent of v times the negative sine of u
dy by dx equals 2 times (the exponent of (-2x)) times (the sine of the exponent of (-2x))

2 Differentiate:

y equals the natural logarithm of the sine of (1 minus 3x)
y equals the natural logarithm of u and u equals the sine of (1 minus 3x)
u equals the sine of v and v = 1 minus 3x
dy by du equals 1 over u, du by dv equals the cosine of v and dv by dx equals -3
dy by dx equals dy by du times du by dv times dv by dx equals (1 over u) times the cosine of v times -3
dy by dx equals (-3) times (1 over the sine of (1 minus 3x)) times (the cosine of (1 minus 3x) which equals (-3) times the cotangent of (1 minus 3x)

3 Express this function in one of the usual forms then differentiate:

y equals the square root of ((4 times the natural logarithm of (x plus 1) ) raised to the power 5) which equals (4 times the natural logarithm of (x plus 1) ) raised to the power (5 over 2)
y equals u to the power of (5 over 2) and u equals 4 times the natural logarithm of (x plus 1)
u equals 4 times the natural logarithm of v and v equals x plus 1
dy by du equals (5 over2) times (u to the power (3 over 2)), du by dv equals 4 over v and dv by dx equals 1
dy by dx equals dy by du times du by dv times dv by dx equals (5 over 2) times (u to the power (3 over 2)) times (4 over v) times 1
dy by dx equals (5 over 2) times ((4 times the natural logarithm of (x plus 1)) to the power (3 over 2)) times (4 over (x plus 1)) times 1 which equals (10 times the square root of (4 times the natural logarithm of (x plus 1)) cubed) over (x plus 1)

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