Definitions [2]
Definition: Composite Function
If \(u = g(x)\) and \(y = f(u)\), then \(y = f(g(x))\) is called a composite function. Here, \(g(x)\) is the inner function and \(f(u)\) is the outer function.
Definition: Chain Rule
Let \[f\] be a real-valued function which is a composite of two functions \[u\] and \[v\]; i.e., \[f = v \circ u\]. Suppose \[t = u(x)\] and if both \[\frac{dt}{dx}\]and \[\frac{dv}{dt}\]exist, we have
\[\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}\]
Formulae [1]
Formula: Derivative of Composite Functions
| Function | Derivative |
|---|---|
| [f(x)]ⁿ | n[f(x)]ⁿ⁻¹ · f′(x) |
| \[\sqrt{\mathrm{f}(x)}\] | \[\frac{1}{2\sqrt{\mathrm{f}(x)}}\cdot\mathrm{f}^{\prime}(x)\] |
| \[\frac{1}{\mathrm{f}(x)}\] | \[-\frac{1}{\left[\mathrm{f}(x)\right]^{2}}\cdot\mathrm{f}^{\prime}(x)\] |
| sin(f(x)) | cos(f(x)) · f′(x) |
| cos(f(x)) | −sin(f(x)) · f′(x) |
| tan(f(x)) | sec²(f(x)) · f′(x) |
| cot(f(x)) | −cosec²(f(x)) · f′(x) |
| sec(f(x)) | sec(f(x)) tan(f(x)) · f′(x) |
| cosec(f(x)) | −cosec(f(x)) cot(f(x)) · f′(x) |
| \[\mathbf{a}^{\mathbf{f}(x)}\] | \[a^{f(x)}\log a\cdot f^{\prime}(x)\] |
| \[\mathrm{e}^{\mathrm{f}(x)}\] | \[\mathrm{e}^{\mathrm{f}(x)\cdot\mathrm{f}^{\prime}(x)}\] |
| log(f(x)) | \[\frac{1}{\mathrm{f}(x)}\cdot\mathrm{f}^{\prime}(x)\] |
| logₐ(f(x)) | \[\frac{1}{\mathrm{f}(x)\mathrm{loga}}\cdot\mathrm{f}^{\prime}(x)\] |
Key Points
Key Points: Derivative of Composite Functions
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A composite function has one function inside another function.
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The chain rule formula is \[\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}\]
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First differentiate the outer function, then multiply by the derivative of the inner function.
Important Questions [13]
- F Z = Tan ( Y − a X ) + ( Y − a X ) 3 2 Then Show that ∂ 2 Z ∂ X 2 = a 2 ∂ 2 Z ∂ Y 2
- If U = E X Y Z F ( X Y Z ) Where F ( X Y Z ) is an Arbitrary Function of X Y Z . Prove That: X ∂ U ∂ X + Z ∂ U ∂ Z = Y ∂ U ∂ Y + Z ∂ U ∂ Z = 2 X Y Z . U
- Find the Maximum and Minimum Values of F ( X , Y ) = X 3 + 3 X Y 2 − 15 X 2 − 15 Y 2 + 72 X
- If Z=F(X.Y). X=R Cos θ, Y=R Sinθ. Prove that ( ∂ Z ∂ X ) 2 + ( ∂ Z ∂ Y ) 2 = ( ∂ Z ∂ R ) 2 + 1 R 2 ( ∂ Z ∂ θ ) 2
- If U = F ( Y − X X Y , Z − X X Z ) , Show that X 2 ∂ U ∂ X + Y 2 ∂ U ∂ Y + Z 2 ∂ U ∂ Z = 0 .
- If U = Sin − 1 ( X + Y √ X + √ Y ) ,Prove that X 2 U X X + 2 X Y U X Y + Y 2 U Y Y = − Sin U . Cos 2 U 4 Cos 3 U
- State and Prove Euler’S Theorem for Three Variables.
- State and Prove Euler’S Theorem for Three Variables.
- If Z = F (X, Y) Where X = Eu +E-v, Y = E-u - Ev Then Prove that ∂ Z ∂ U − ∂ Z ∂ V = X ∂ Z ∂ X − Y ∂ Z ∂ Y .
- If U = Sin − 1 [ X 1 3 + Y 1 3 X 1 2 + Y 1 2 ] Prove Hat X 2 ∂ 2 U ∂ 2 X + 2 X Y ∂ 2 U ∂ X ∂ Y ∂ 2 U ∂ 2 Y = Tan U 144 [ Tan 2 U + 13 ] .
- If x = u+v+w, y = uv+vw+uw, z = uvw and φ is a function of x, y and z Prove that
- If Tan(θ+Iφ)=Tanα+Isecα Prove that 1) E 2 φ = Cot ( φ 2 ) 2) 2 θ = N π + π 2 + α
- State Euler’S Theorem on Homogeneous Function of Two Variables and If U = X + Y X 2 + Y 2 Then Evaluate X ∂ U ∂ X + Y ∂ U ∂ Y
