Find the derivative of sin (x2-1)
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Find the derivative of sin(x
2
+1) with respect to x from first principle. (IIT-JEE, 1978)
Medium
Solution
verified
Verified by Toppr
Let F(x)=sin(x
2
+1),then,f(x+h)=sin[(x+h)
2
+1]
∴
h→0
lim
h
f(x+h)−f(x)
=
k→0
lim
h
sin[(x+h)
2
+1]−sin[x
2
+1]
⇒f
′
(x)=
h→0
lim
2cos(
2
2x
2
+h
2
+2xh+2
)×
h
sin(
2
h
2
+2xh
)
=2cos(x
2
+1)
h→0
lim
h[
2
h+2x
]
sin[
2
h
2
+2xh
]
(
2
h+2x
)
=2xcos(x
2
+1)
Derivative of sin(x² - 1).
let the function, y = sin(x² - 1)
Differentiate on both the sides wrt x :
→ dy/dx = d(sin(x² - 1))/dx
Apply chain rule,
dy/dx = dy/du × du/dx
Here, put u = (x² - 1)
=> dy/dx = d(sin(x² - 1))/d(x² - 1) × d(x² - 1)/dx
=> dy/dx = cos(x² - 1)(2x)
Hence, the derivative of sin(x² - 1) is 2xcos(x² - 1).
Simple!