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You are watching: Differentiate the function. y = tan(ln(ax + b))

y is a composite function in the form `y=f(g(x))` with `f(X)=tan(X)` and `g(x)=ln(ax+b)` .

`f"(X)=1+tan^2X`

`g"(x)=a/(ax+b)`

then` y"=g"(x)*f"(g(x))=a/(ax+b)*(1+tan^2(ln(ax+b)))`

The derivative is

`y"(x)=a(1+tan^2(ln(ax+b)))/(ax+b)`

Remark. There is a second expression for the derivative of tan X:

`tan"X=1/cos^2X`

It is the same function but written with...

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y is a composite function in the form `y=f(g(x))` with `f(X)=tan(X)` and `g(x)=ln(ax+b)` .

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`f"(X)=1+tan^2X`

`g"(x)=a/(ax+b)`

then` y"=g"(x)*f"(g(x))=a/(ax+b)*(1+tan^2(ln(ax+b)))`

The derivative is

`y"(x)=a(1+tan^2(ln(ax+b)))/(ax+b)`

Remark. There is a second expression for the derivative of tan X:

`tan"X=1/cos^2X`

It is the same function but written with a different expression. Using this expression in the derivative of the composite fonction we find a different expression of the same derivative:

`y"(x)=a/((ax+b)cos^2(ln(ax+b)))`

` `

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