Differentiation of function of function

If y = f(u) and u = g(x), then dy/dx = (dy/du).(du/dx) = f'(u) . g'(x) = f'(g(x)) . g’ (x)

Illustration : Find dy/dx

(i)

(ii)

(iii)

Solution: (i)

Differentiation of implicit functions

Given the equation f(x,y) = c

For finding dy/dx , we differentiate both the sides w.r.t, x, treating y as a function of x , and then solve equation for dy/dx.

Illustration : Find dy/dx

(i) log(xy) = x2 + y2

(ii) x+y = sin(xy)

Solution: (i) log(xy) = x2 + y2

=> logx + logy = x2 + y2

Differentiating w. r. to x ,

(ii) x + y = sin(xy)

Differentiating w.r.t. x, we get

=> ( dy/dx)[1 − x cos (xy)] = y cos (xy) − 1

=

Differentiation of parametric functions

Working Rule

(i) If x and y are function of parameter t , then find dx/dt and dy/dt separately.

(ii) Then

e.g., x = a(θ + sinθ), y = a(1 − cosθ) where θ is parameter.

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