Composite, Implicit, and Inverse Functions

Using chain rule successively for each layer starting from inside out.

Taking advantage of implicit differentiation treating each layer as an implicit variable.

Applying product rule on split parts of the composite function.

Integrating each individual function before applying differentiation rules.

108

12

75

24

9cos(x)

sin(9)

9cos(9x-8)

−9cos(8−9x)

No, it's not a composite function

Yes, outer function is $x^2+5x+2$ and inner function is $x^5$

Yes, they both have same inner and outer function $x^n$

Yes, inner function is $x^2+5x+2$ and outer function is $x^5$

Yes, inner function is $\ln(x)$ and outer function is $\sin(x)$

Yes, they both have same inner and outer function $f(x)$

Yes, inner function is $\sin(x)$ and outer function is $\ln(x)$

No, it's not a composite function

Each function (u,v,w,and z) should be differentiated separately without compounding other derivatives.

$f'(u) \cdot g''(v(w(z))) \cdot v'(w(z)) \cdot w'(z)$

Threat each function (u,v,w,and z) as separate entities when calculating the chain rule.

Consider only derivatives of f(u) and d(u)/dz, ignoring intermediate layers of formulation.

$(3t^2 \tan(t) + t^3 \sec^2(t))$

$(6t \tan(t) + t^{-4} \sec^{-4}(t))$

$(t^{-3} \sec^{-3}(t))$

$(3t^{-3} \tan(t))$

$(8x+3)\\sin(4x^2+3x)$

$\\cos(4x^2+3x)$

$(8x+3)\\cos(4x^2+3x)$

$\\cos(8x+3)$

$-30x^{10}$

$-3 imes 10^x$

$-3 imes \\ln(10) imes 10^x$

$-3 imes \\log(x) imes 10^x$

Function Evaluation and Function Composition.

Commutation, Addition, Multiplication.

Product Rule and Chain Rule.

Differential and Integration.