Calculus Examples

Find the Derivative - d/dx y=csc(x)^2
y=csc2(x)
Step 1
Differentiate using the chain rule, which states that ddx[f(g(x))] is f(g(x))g(x) where f(x)=x2 and g(x)=csc(x).
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Step 1.1
To apply the Chain Rule, set u as csc(x).
ddu[u2]ddx[csc(x)]
Step 1.2
Differentiate using the Power Rule which states that ddu[un] is nun-1 where n=2.
2uddx[csc(x)]
Step 1.3
Replace all occurrences of u with csc(x).
2csc(x)ddx[csc(x)]
2csc(x)ddx[csc(x)]
Step 2
The derivative of csc(x) with respect to x is -csc(x)cot(x).
2csc(x)(-csc(x)cot(x))
Step 3
Multiply -1 by 2.
-2csc(x)(csc(x)cot(x))
Step 4
Raise csc(x) to the power of 1.
-2(csc1(x)csc(x))cot(x)
Step 5
Raise csc(x) to the power of 1.
-2(csc1(x)csc1(x))cot(x)
Step 6
Use the power rule aman=am+n to combine exponents.
-2csc(x)1+1cot(x)
Step 7
Add 1 and 1.
-2csc2(x)cot(x)
y=csc2(x)
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 [x2  12  π  xdx ]