# Solve Derivative: log 5^xsinhx

• ldbaseball16
In summary, Unco pointed out that you may have an exponential function raised to a power, but it is not a product. You should differentiate (ln 5)* x*sinh(x) and get two terms.
ldbaseball16

## Homework Statement

find f '(x) of 5^xsinhx

## Homework Equations

i feel like I am missing somthing can someone direct me to the right direction pleasee?

## The Attempt at a Solution

product rule (f ')(g)+(f)(g ') f=5^x g=sinhx

5^xsinhx= e^ln(5)^xsinhx

(e^ln(5)^x)(sinhx)+(5^x)(coshx)(ln5)

(e^ln(5)^x)(sinhx)+(5^x)(ln5)(coshx)?

Well it looks like you took the derivative of none of the terms in the first term, and both of the terms in the second term. Try to fix that.

ldbaseball16 said:

## Homework Statement

find f '(x) of 5^xsinhx

## Homework Equations

i feel like I am missing somthing can someone direct me to the right direction pleasee?

## The Attempt at a Solution

product rule (f ')(g)+(f)(g ') f=5^x g=sinhx

5^xsinhx= e^ln(5)^xsinhx

(e^ln(5)^x)(sinhx)+(5^x)(coshx)(ln5)

(e^ln(5)^x)(sinhx)+(5^x)(ln5)(coshx)?
The factor of ln(5) is in the wrong term.

Write them out separately:
f =
f' =
g =
g' =

Then put them together.

By the way, ensure you use parentheses to remove ambiguity when typing exponents; e.g., 5^x = (e^(ln(5)))^x = $$\left(e^{\ln{5}}\right)^x$$ etc.

"none of the terms in the first term" what do you mean by this?

ldbaseball16 said:

## Homework Statement

find f '(x) of 5^xsinhx
As Unco pointed out, your expression above is ambiguous. It's not clear what the exponent on 5 is. It could be x, or it could be x*sinh(x). In other words, is this function (5^x)* sinh(x) or is it (5^x)^sinh(x)?
ldbaseball16 said:

## Homework Equations

i feel like I am missing somthing can someone direct me to the right direction pleasee?

## The Attempt at a Solution

product rule (f ')(g)+(f)(g ') f=5^x g=sinhx
Now from this work it appears that the function is (5^x) * sinh(x), in other words, as the product of two functions. Inasmuch as you're using the product rule, you must think that the function is (5^x)*sinh(x).
ldbaseball16 said:
5^xsinhx= e^ln(5)^xsinhx
Now it looks like an exponential function raised to a power, so the function is not a product, which means that the product rule does not apply.

What exactly is the function? We can't help you if you don't know what it is.
ldbaseball16 said:
(e^ln(5)^x)(sinhx)+(5^x)(coshx)(ln5)

(e^ln(5)^x)(sinhx)+(5^x)(ln5)(coshx)?

5^x is NOT "e^ln(5)^x" it is e^(x ln(5))

Which means that (5^x)'= (e^(x ln(5)))'= ln(5) e^(x ln(5))= ln(5) 5^x.

here is the original problem find f '(x).

f(x)=5^xSinhx does this clarify?

oooh, why can't everyone learn how to use the X2 tag (just above the Reply box)?

It's e(ln5)xsinhx.​

yea and that's what i put on top? so since the product rule is (f ')(g)+(f)(g ')
so wouldn't the answer be (e^ln(5)xsinhx+(5^x)(coshx)?

ldbaseball16 said:
yea and that's what i put on top?

nooo! try the X2 tag!
so since the product rule is (f ')(g)+(f)(g ')
so wouldn't the answer be (e^ln(5)xsinhx+(5^x)(coshx)?

product rule?

what product?

use the chain rule first!

i don't think its the right answer e^sinhx(coshx)

Last edited:
ok so i got 5^xcoshx+5^x(ln5)sinhx?

ldbaseball16 said:
ok so i got 5^xcoshx+5^x(ln5)sinhx?

not even close

show us your full working …

differentiate e(ln5)xsinhx using the chain rule

i used the chain rule and i got e^(ln5xsinhx)(ln5)??

what is the derivative of (ln5)xsinhx?

1/5xcoshx

It looks like you are doing this: d/dx(ln 5) = 1/5

If so, that's wrong.

Also, when you differentiate (ln 5)* x*sinh(x), you should be getting two terms.

The answer in your previous post might be interpreted in several ways:
$$\frac{1}{5x cosh(x)}$$
$$\frac{1}{5x}cosh(x)$$
$$\frac{1}{5}xcosh(x)$$

Use parentheses or learn LaTex!

ln5*sinh(x)+ln5*x*cosh(x)?

Yes, that's it. Now you can respond to Tiny Tim's request in post 13.

## 1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.

## 2. How do I solve a derivative?

To solve a derivative, you need to use the rules of differentiation, which include the power rule, product rule, quotient rule, and chain rule. You also need to have a good understanding of algebra and trigonometry.

## 3. What is the derivative of log 5^xsinhx?

The derivative of log 5^xsinhx can be solved using the product rule and the chain rule. The final answer is x(log 5)sinhx + log 5^x (coshx).

## 4. Why is the derivative important?

The derivative is important because it allows us to determine the rate of change of a function, which has many real-world applications in fields such as physics, engineering, economics, and more. It also helps us find critical points, which can be used to optimize functions.

## 5. Can you explain the concept of logarithmic differentiation?

Logarithmic differentiation is a method used to solve derivatives of functions that are not easily solved using traditional rules. It involves taking the natural logarithm of both sides of the function and then differentiating. This method is particularly useful for functions that involve products, quotients, or powers.

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