Could someone please explain this to me?

Question

Could someone please explain this to me?

Differentiate.

h(u)= 10^(sqrt(u))

h'(u)= 10^(sqrt(u))*log(10)(d/du(sqrt(u)))

h'(u)= 10^(sqrt(u))*((1)/(2*sqrt(u)))*log(10)

Answer 1

let x = √u = u^(1/2) , let h(u) = y

then y = 10^x
take ln of both sides
ln y = ln 10^x
ln y = xln10
(dy/dx) / y= ln10
dy/dx = y ln10
= (10^x)(ln10)

back to x = u^(1/2)
dx/du = (1/2)u^(-1/2)

then (dy/dx) (dx/du) = [(10^x)(ln10)(1/2)(u^(-1/2)
dy/du = (10^√u)(ln10)/(2√u)

you have log10, it should be ln10

Answer 2

Note that the "log(10)" above refers to natural log of 10.

There was a time before calculators when log₁₀(x) was used to help do multiply/divide operations. To avoid confusion, natural log was denoted by ln(x). It has stayed since.

However, log(x) continues to be used for natural log in higher mathematics.

Start with
y=10^x
take log₁₀ on both sides:
log₁₀y=x
differentiate:
(1/(y*ln(10)))(dy/dx)=1
dy/dx=y*ln(10)
dy/dx = 10^x ln(10)

The rest is just the chain rule, for example,
y=sin(x)^4
dy/dx = 4sin(x)^3*d(sin(x))/dx
=4sin(x)^3 * cos(x)

Answer 3

To differentiate the function h(u) = 10^(sqrt(u)), you can use the chain rule.

First, let's differentiate the outer function, which is 10^(sqrt(u)). To differentiate any function of the form a^b, where a is a constant and b is a function of u, you need to multiply it by a^b * log(a) * (d/du)b.

In this case, a = 10 and b = sqrt(u). So the derivative of 10^(sqrt(u)) is:

h'(u) = 10^(sqrt(u)) * log(10) * (d/du)sqrt(u)

Next, let's differentiate the inner function, sqrt(u), with respect to u.

To differentiate sqrt(u) with respect to u, you can use the chain rule again. The derivative of sqrt(u) is (1/2sqrt(u)) * (d/du)u.

Putting it all together, the derivative of h(u) = 10^(sqrt(u)) is:

h'(u) = 10^(sqrt(u)) * log(10) * (1/2sqrt(u)) * (d/du)u

Simplifying further, we get:

h'(u) = 10^(sqrt(u)) * (1/2sqrt(u)) * log(10)

Answer 4

To differentiate the function h(u) = 10^(sqrt(u)), we can use the chain rule, which states that if we have a composite function such as f(g(u)), then the derivative of that function with respect to u is f'(g(u)) * g'(u).

In this case, let's consider the function g(u) = sqrt(u). To find the derivative of g(u) with respect to u, we can apply the power rule for derivatives. The power rule states that if we have a function f(x) = x^n, then the derivative of f(x) with respect to x is f'(x) = n * x^(n-1).

Using the power rule, we find that g'(u) = (1/2) * u^(-1/2).

Now, let's apply the chain rule to find the derivative of h(u). We have h(u) = 10^(sqrt(u)), where f(x) = 10^x and g(u) = sqrt(u).

By applying the chain rule, we get h'(u) = f'(g(u)) * g'(u).

From our calculations, we know that f'(x) = log(10)(10) = 1 since log(10)(10) is the base-10 logarithm of 10, which equals 1.

Therefore, h'(u) = 1 * (10^(sqrt(u))) * [(1/2) * u^(-1/2)].

Simplifying this expression, we can write:

h'(u) = 10^(sqrt(u)) * [(1/2) * u^(-1/2)] * log(10)(10).

Since log(10)(10) is equal to 1, our final expression becomes:

h'(u) = 10^(sqrt(u)) * [(1/2) * u^(-1/2)].

And that's how you differentiate the function h(u) = 10^(sqrt(u)) using the chain rule.