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Differentiate the functions with respect to x.
Let f(x)=cos(sinx),
u(x)=sinx , v(t)=cost
Where t=u(x)=sinx
Differentiating it
Since. By chain rule,
Differentiate the functions with respect to x.
Let f(x)=sin(x2+5), u(x)=x2+5, and v(t)=sintLet f(x)=sinx2+5, ux=x2+5, and v(t)=sint
Then, (vou)=v(u(x))=v(x2+5)=tan(x2+5)=f(x)Then, vou=vux=vx2+5=tanx2+5=f(x)
Thus, f is a composite of two functions.
Alternate method
Differentiate the functions with respect to x.
Thus, f is a composite function of two functions, u and v.
Put t = u (x) = ax + b
Hence, by chain rule, we obtain
Alternate method
Differentiate the functions with respect to x.
The given function is
, where g (x) = sin (ax + b) and
h (x) = cos (cx + d)
∴ g is a composite function of two functions, u and v.
Therefore, by chain rule, we obtain
∴h is a composite function of two functions, p and q.
Put y = p (x) = cx + d
Therefore, by chain rule, we obtain
Differentiate the functions with respect to x.
Clearly, f is a composite function of two functions, u and v, such that
By using chain rule, we obtain
Alternate method
Prove that the function f given by
is notdifferentiable at x = 1
The given function is
It is known that a function f is differentiable at a point x = c in its domain if both
are finite and equal.
To check the differentiability of the given function at x = 1,
consider the left hand limit of f at x = 1
Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1
Prove that the greatest integer function defined by
is not
differentiable at x = 1 and x = 2.
The given function f is
It is known that a function f is differentiable at a point x = c in its domain if both
are finite and equal.
To check the differentiability of the given function at x = 1, consider the left hand limit of f at x = 1
Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at
x = 1
To check the differentiability of the given function at x = 2, consider the left hand limit
of f at x = 2
Since the left and right hand limits of f at x = 2 are not equal, f is not differentiable at x = 2
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