Jawairia's questions at Yahoo! Answers regarding a tangent line and logarithmic differentiation

[MarkFL]

Here are the questions:

Find eq of tangent and logarithm differentiation....?

Find an equation of the tangent line to the curve
y=(x^4 -3x^2+2x) * (x^3-2x+3) at x = 0.


b:
Use the logarithmic differentiation to differentiate the function

F(x)=(2x+1)^2 * (3x^2-4)^7 *(x+7)^4

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello Jawairia,

a) We can save a lot of work by observing we have:

\(\displaystyle y=x^7+\cdots+6x\)

and so, we will find:

\(\displaystyle y'(0)=6\)

We can also easily see that:

\(\displaystyle y(0)=0\)

and so the tangent line must be:

\(\displaystyle y-0=6(x-0)\)

\(\displaystyle y=6x\)

b) We are given:

\(\displaystyle F(x)=(2x+1)^2(3x^2-4)^7(x+7)^4\)

If we take the natural log of both sides, we obtain:

\(\displaystyle \ln(F(x))=\ln\left((2x+1)^2(3x^2-4)^7(x+7)^4 \right)\)

Applying the properties of logarithms, we may write:

\(\displaystyle \ln(F(x))=2\ln(2x+1)+7\ln\left(3x^2-4 \right)+4\ln(x+7)\)

Differentiating with respect to $x$, we find:

\(\displaystyle \frac{1}{F(x)}F'(x)=\frac{4}{2x+1}+\frac{42x}{3x^2-4}+\frac{4}{x+7}\)

Hence:

\(\displaystyle F'(x)=F(x)\left(\frac{4}{2x+1}+\frac{42x}{3x^2-4}+\frac{4}{x+7} \right)\)

Replacing $F(x)$ with its definition, we find:

\(\displaystyle F'(x)=(2x+1)^2(3x^2-4)^7(x+7)^4\left(\frac{4}{2x+1}+\frac{42x}{3x^2-4}+\frac{4}{x+7} \right)\)

Distributing, we get:

\(\displaystyle F'(x)=4(2x+1)(3x^2-4)^7(x+7)^4+42x(2x+1)^2(3x^2-4)^6(x+7)^4+4(2x+1)^2(3x^2-4)^7(x+7)^3\)

Factoring, we find:

\(\displaystyle F'(x)=2(2x+1)(3x^2-4)^6(x+7)^3\left(2(3x^2-4)(x+7)+21x(2x+1)(x+7)+2(2x+1)(3x^2-4) \right)\)

Expanding and collecting like terms, we finally find:

\(\displaystyle F'(x)=2(2x+1)(3x^2-4)^6(x+7)^3\left(60x^3+363x^2+123x-64 \right)\)