Question

if y=f(x^3) and f'(x)=e^5x, find dy/dx

please answer asap !​

Answer 1

Step-by-step explanation:

${f}^{1} (x) = {e}^{5x} \\ \displaystyle \int {f}^{1} (x) \: dx = \displaystyle \int \: {e}^{5x} dx \\f(x) \: = \dfrac{ {e}^{5x} }{5} + c \\ y = f( {x}^{3} ) \\ y = \dfrac{ {e}^{5 {x}^{3} } }{ 5} + c \\ differentiating \: with \: respect \: to \: x \\ \frac{dy}{dx} = \dfrac{1}{5} {e}^{5 {x}^{3} } \frac{d}{dx} (5 {x}^{3} ) + 0 \\ = \dfrac{1}{5} {e}^{5 {x}^{3} } (15 {x}^{2} ) \\ = 3 {e}^{5 {x}^{3} } {x}^{2}$

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:y = f( {x}^{3})$

and

$\rm :\longmapsto\:f'(x) = {e}^{5x}$

Now, As

$\rm :\longmapsto\:f'(x) = {e}^{5x}$

$\bf\implies \:\:f'( {x}^{3} ) = {e}^{5 {x}^{3} } - - - - (1)$

Now, Consider

$\rm :\longmapsto\:y = f( {x}^{3})$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx} y =\dfrac{d}{dx} f( {x}^{3})$

$\rm :\longmapsto\:\dfrac{dy}{dx} = f'( {x}^{3})\dfrac{d}{dx} {x}^{3}$

We know,

$\boxed{ \tt{ \: \dfrac{d}{dx} {x}^{n} = {x}^{n - 1} \: }}$

So, using this, we get

$\rm :\longmapsto\:\dfrac{dy}{dx} = f'( {x}^{3}) \times 3{x}^{2}$

$\rm :\longmapsto\:\dfrac{dy}{dx} = 3{x}^{2} {e}^{ {5x}^{3} } \: \: \: \: \: \: \{using \: (1) \}$

Hence,

$\rm :\longmapsto\:\boxed{ \tt{ \: \dfrac{dy}{dx} = 3{x}^{2} {e}^{ {5x}^{3} } \: }}$

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