Taking derivative of this equation

Question

$(a+1)^4 \cdot 0.46^a \cdot 0.58^a \cdot 0.71^a \cdot 0.92^a$

I have difficulties with taking the derivative of this function above without using graphic calculator. Someone who knows how to do it?

Answer 1

Let $y = (a+1)^4 \cdot 0.46^a \cdot 0.58^a \cdot 0.71^a \cdot 0.92^a$. You can rewrite this as: $y = (a+1)^4 \cdot 0.17427376 ^a$, since $n^a \cdot m^a =(n \cdot m)^a$.

Now, you can use the product rule to evaluate the derivative:

$y' = \left[(a+1)^4 \right]'\cdot 0.17427376 ^a +(a+1)^4 \cdot \left[0.17427376 ^a \right]'$

Using the chain rule we know that $\left[(a+1)^4 \right]' = 4(a + 1)^3$ and,

$$\begin{align} \left[0.17427376^a \right]'&=\left[e^{a \ln 0.17427376} \right]' \\ &=\ln 0.17427376 \cdot e^{a \ln 0.17427376} \\ &= \ln 0.17427376 \cdot 0.17427376^a \end{align} $$

So you final answer for the derivative is:

$y'= 4(a + 1)^3\cdot 0.17427376 ^a +(a+1)^4 \cdot \ln 0.17427376 \cdot 0.17427376^a$

If you wish you can simplify the expression you can write:

$y'= \left(4(a + 1)^3 +(a+1)^4 \cdot \ln 0.17427376 \right)\cdot 0.17427376^a$

This is you derivative