How to get derivative of integral with $ln(x)$?
$begingroup$
Find $f'(x)$ if $f(x) = int_1^{ln(x)} e^{t^2} ,dt$
The correct way to solve it:
$$f'(x) = e^{(ln{x})^2} frac{1}{x}$$
$$f'(x) = frac{1}{x}e^{(ln{x})^2}$$
I haven't seen an example like this before, but it seems relatively easy to solve.
I'm assuming the exponent $t$ gets replaced with $ln x$ and the $frac{1}{x}$ comes from the derivative of $ln x$ via the chain rule? But what are the exact steps to take to get the derivative of this integral?
calculus integration derivatives definite-integrals
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
$endgroup$
add a comment |
$begingroup$
Find $f'(x)$ if $f(x) = int_1^{ln(x)} e^{t^2} ,dt$
The correct way to solve it:
$$f'(x) = e^{(ln{x})^2} frac{1}{x}$$
$$f'(x) = frac{1}{x}e^{(ln{x})^2}$$
I haven't seen an example like this before, but it seems relatively easy to solve.
I'm assuming the exponent $t$ gets replaced with $ln x$ and the $frac{1}{x}$ comes from the derivative of $ln x$ via the chain rule? But what are the exact steps to take to get the derivative of this integral?
calculus integration derivatives definite-integrals
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
$endgroup$
3
$begingroup$
That's correct, and this is just a one-line application of the fundamental theorem of calculus. If $F$ is the antiderivative of $f$, you're just being asked to differentiate the function $x mapsto F(ln x)$.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:19
add a comment |
$begingroup$
Find $f'(x)$ if $f(x) = int_1^{ln(x)} e^{t^2} ,dt$
The correct way to solve it:
$$f'(x) = e^{(ln{x})^2} frac{1}{x}$$
$$f'(x) = frac{1}{x}e^{(ln{x})^2}$$
I haven't seen an example like this before, but it seems relatively easy to solve.
I'm assuming the exponent $t$ gets replaced with $ln x$ and the $frac{1}{x}$ comes from the derivative of $ln x$ via the chain rule? But what are the exact steps to take to get the derivative of this integral?
calculus integration derivatives definite-integrals
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
$endgroup$
Find $f'(x)$ if $f(x) = int_1^{ln(x)} e^{t^2} ,dt$
The correct way to solve it:
$$f'(x) = e^{(ln{x})^2} frac{1}{x}$$
$$f'(x) = frac{1}{x}e^{(ln{x})^2}$$
I haven't seen an example like this before, but it seems relatively easy to solve.
I'm assuming the exponent $t$ gets replaced with $ln x$ and the $frac{1}{x}$ comes from the derivative of $ln x$ via the chain rule? But what are the exact steps to take to get the derivative of this integral?
calculus integration derivatives definite-integrals
calculus integration derivatives definite-integrals
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
asked Dec 11 '18 at 0:15
qbufferqbuffer
84
84
3
$begingroup$
That's correct, and this is just a one-line application of the fundamental theorem of calculus. If $F$ is the antiderivative of $f$, you're just being asked to differentiate the function $x mapsto F(ln x)$.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:19
add a comment |
3
$begingroup$
That's correct, and this is just a one-line application of the fundamental theorem of calculus. If $F$ is the antiderivative of $f$, you're just being asked to differentiate the function $x mapsto F(ln x)$.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:19
3
3
$begingroup$
That's correct, and this is just a one-line application of the fundamental theorem of calculus. If $F$ is the antiderivative of $f$, you're just being asked to differentiate the function $x mapsto F(ln x)$.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:19
$begingroup$
That's correct, and this is just a one-line application of the fundamental theorem of calculus. If $F$ is the antiderivative of $f$, you're just being asked to differentiate the function $x mapsto F(ln x)$.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:19
add a comment |
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$begingroup$
That's correct, and this is just a one-line application of the fundamental theorem of calculus. If $F$ is the antiderivative of $f$, you're just being asked to differentiate the function $x mapsto F(ln x)$.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:19